In Copyright - Non-Commercial Use Permitted Rights ...23817/eth-23817-02.pdfAbstract Thisworkis devotedto hypergeometricseries. In the first chapter, weconstruct explicitely abelian

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Doctoral Thesis

Abelian varieties and identities for hypergeometric series

Author(s): Archinard, Natália

Publication Date: 2000

Permanent Link: https://doi.org/10.3929/ethz-a-004070858

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Diss.ETHNo. 13882

Abelian Varieties and Identities

for Hypergeometric Series

A dissertation submitted to the

Swiss Federal Institute of Technology Zurich

for the degree of

Doctor of Mathematics

presented by

Natalia Archinard

Dipl. Math. Univ. Genève

born on the 11th of June 1973

citizen of Geneva (GE)

accepted on the recommendation of

Prof. Dr. Gisbert Wüstholz, examiner

Prof. Dr. Paula B. Cohen, co-examiner

2000

Abstract

This work is devoted to hypergeometric series.

In the first chapter, we construct explicitely abelian varieties associated to Appell-Lauricella hypergeometric series. From our general construction, we deduce the

constructions of abelian varieties for Gauss' hypergeometric series and for Beta-

functions.

In the second chapter, we produces identities involving one Gauss' hypergeomet¬ric series on one hand and, on the other hand, the modular J-invariant and the

Dedekind ^-function. With such identities, we are able to show the algebraicity of

some Gauss' hypergeometric series on some infinite subset of Q and to calculate ex¬

plicitely the algebraic value of F(j^, -p-, |; z) at some points of the correspondingsubset.

Zusammenfassung

Diese Arbeit ist hypergeometrischen Reihen gewidmet.

Im ersten Kapitel bauen wir abelsche Varietäten, die in engem Zusammenhangmit Appell-Lauricella hypergeometrischen Reihen stehen. Als spezielle Fälle un¬

serer allgemeinen Konstruktion erhalten wir abelsche Varietäten einerseits für die

Gauss'sche hypergeometrische Reihe und anderseits für die Beta-Funktion.

Im zweiten Kapitel beweisen wir verschiedene Identitäten. Jede Identität bringteine Gauss'sche hypergeometrische Reihe in Verbindung mit der /-Modulfunktion

und der Dedekind'sehen ^-Funktion. Mit Hilfe solcher Identitäten zeigen wir in

gewissen Fällen, dass die entsprechende hypergeometrische Reihe einen algebrais¬chen Wert in jedem Punkt einer gewissen unendlichen Untermenge von Q an¬

nimmt. Ferner berechnen wir explizit ein paar algebraische Auswertungen von

F{y^., y^, ^; z) fût z aus der entsprechenden Untermenge.

Contents

Introduction 1

1 Abelian Varieties Associated to Hypergeometric Series 5

1.1 Definitions and Basic Facts on Hypergeometric Series 5

1.2 Motivation and Structure of the Chapter 8

1.3 Preparatory Material 9

1.3.1 Affine Algebraic Varieties 9

1.3.2 Projective Algebraic Varieties 10

1.3.3 Irreducibility and Dimension 11

1.3.4 Algebraic Curves 12

1.3.5 Morphisms and Birational Equivalence 13

1.3.6 Blow-up 14

1.3.7 Desingularization of a Plane Curve 16

1.3.8 Riemann Surfaces and Nonsingular AlgebraicCurves over C 18

1.3.9 Topological Invariants of a Riemann Surface 22

i

11 Contents

1.3.10 Tangent Space to an Algebraic Variety and Differential Maps 28

1.3.11 Differential Forms on Algebraic Varieties 33

1.3.12 Differential Forms on Riemann Surfaces 36

1.3.13 Integration of Holomorphic Differential 1-Forms on Rie¬

mann Surfaces 39

1.3.14 Sheaves 42

1.3.15 Complex Tori and Abelian Varieties over C 43

1.3.16 Jacobian Varieties over C 46

1.3.17 Algebraic Appendix 49

1.4 Abelian Varieties for Appell-Lauricella Hypergeometric Series...

53

1.4.1 Motivation and Structure of the Section 53

1.4.2 Reduction 54

1.4.3 Definition of the Family of Curves 56

1.4.4 Singular Points on Ca? 57

1.4.5 Structure of the Desingularization 57

1.4.6 Genus of XN 67

1.4.7 Actions of/x/y 71

1.4.8 Basis of Regular Differential Forms on Xj\r 73

1.4.9 New Forms 83

1.4.10 New Jacobian 85

1.5 Abelian Varieties for Gauss'Hypergeometric Series 86

1.5.1 Motivation 86

Contents ni

1.5.2 The Isomorphism 88

1.5.3 Genus and New Jacobian of X(N,z) 89

1.6 Abelian Varieties for the Beta Function 92

1.6.1 Motivation 92

1.6.2 The Isomorphism 94

1.6.3 Genus and New Jacobian of X(M, 0) 95

1.7 Degenerated Fibers 97

1.7.1 New Notations and Definitions 97

1.7.2 Dimension of the New Jacobian and Genus of some Fibers.

98

2 Identities for some Gauss' Hypergeometric Series 101

2.1 Introduction 101

2.2 Theoretical Preliminaries 102

2.2.1 Elliptic Curves over C 102

2.2.2 Isomorphisms between Elliptic Curves 104

2.2.3 Change of Basis of a Lattice 104

2.2.4 From a Lattice Function to a Modular one,

Definition of Modular Functions and Forms 105

2.2.5 Eisenstein Series 106

2.2.6 The Discriminant Function A 108

2.2.7 The /-Function 108

2.2.8 The Dedekind rj-Function 109

iv Contents

2.3 The Method of Beukers and Wolfart Ill

2.4 Sixteen Solutions 114

2.4.1 List of Sixteen Solutions 114

2.4.2 Linear Differential Equation of Order Two,

Riemann Scheme and Kummer's 24 Solutions 116

2.4.3 A New Eye on our Solutions 122

2.5 Twenty-two Identities 124

2.5.1 About/= 0 125

2.5.2 About /= 1 131

2.5.3 About / = oo 134

2.6 Some Applications of our Identities 136

2.6.1 Algebraicity of some Hypergeometric Values 136

2.6.2 Some Algebraic Evaluations 137

Acknowledgments 153

Curriculum Vitae 155

Introduction

This work is divided in two parts having the common goal of contributing to the

knowledge on hypergeometric functions. The first chapter constructs tools for the

study of Appell-Lauricella hypergeometric series (these are hypergeometric series

in several variables) and their monodromy groups, while the second chapter is de¬

voted exclusively to Gauss' hypergeometric series (which is the classical hyperge¬ometric series in one variable).

From the time Schwarz [S873] proved that the algebraic Gauss' hypergeometric se¬

ries are exactely those having a finite monodromy group, it has mostly been believed

that no transcendental hypergeometric series could take an algebraic value at an al¬

gebraic point. Surprising enough, Wolfart [Wo85], [W088] showed that the subset

of Q on which the Gauss' hypergeometric series F (a, b, c; z) with a, b, c e Q takes

algebraic values and the complement of this subset are dense in Q exactely when

the monodromy group of F is arithmetic and some conditions on the parameters

a, b, c hold. By Takeuchi [Ta77] and because the monodromy group is triangular,this happens for 85 isomorphy classes of monodromy groups.

The idea in the proof of Wolfart is to interpret Euler's integral representation of

F (a, b, c; z) with a,b, c e Q, z e Q, as a quotient of periods on abelian varieties

defined over Q and to use a consequence ([WW85] Satz 2) of Wiistholz's Analytic

Subgroup Theorem ([Wu89] Hauptsatz) to decide when this quotient can be alge¬braic. To hypergeometric series having the same smallest common denominator

N of a, b, c (hence having SX2(M)-conjugated monodromy group) is associated a

family of abelian varieties of dimension depending on N only. The abelian vari¬

ety associated to an F is an essential factor of the Jacobian variety of the curve on

which the numerator of the integral representation of F is a period (up to an alge¬braic factor). The denominator appears to be (up to an algebraic factor) a periodon a component of the specialization at z = 0 of the above curve. For this case,

1

2 Introduction

the essential factor of the Jacobian had already been constructed and its dimension

calculated by Gross and Rohrlich [GR78].

Later, Cohen and Wolfart [CW90] have used this family of abelian varieties asso¬

ciated to a triangular group A, interpreted as the monodromy group of a hypergeo¬metric differential equation, in order to construct explicitely a modular embeddingof A, that is an embedding into a modular (hence arithmetic) group. The same au¬

thors [CW93] have generalized their work to monodromy groups of hypergeometricfunctions in two variables, defining at this occasion the families of abelian varieties

for hypergeometric series in two variables and giving their dimension.

The families of curves arising from hypergeometric series in several variables have

also been considered by Deligne and Mostow [DM86] in their study of the mon¬

odromy groups of these functions.

In this work, we construct abelian varieties for hypergeometric series in r — 1 vari¬

ables. In order to do this, we first construct explicitely the desingularization of

the singular curve associated to the hypergeometric series, calculate the genus of

the desingularized curve and a basis of holomorphic differential 1-forms on it. We

then define the action of the group of JV-th roots of unity on the vector space of

holomorphic differential 1-forms and calculate the decomposition of this linear rep¬

resentation in irreducible linear subrepresentations as well as the dimension of the

isotypical components. Further, we select some isotypical components. Their direct

sum defines an abelian subvariety of the Jacobian variety of the nonsingular curve.

We calculate the dimension of this subvariety as a function of N (and r) only.

Our results show the coherence of the definitions given in [Wo85], [Wo88], [CW93]and confirm the assertions given inthere about the dimension. On the other hand,

they correct the assertions of [Wo88] about the order of differential 1-forms and

the dimension of the isotypical components. The result on the dimension given in

[GR78] appears as a special case of our dimension's result.

In two cases of Gauss' hypergeometric series with monodromy group isomorphicto SL2(Z), Beukers and Wolfart [BW86] were able to determinate explicitely an

infinite subset of Q on which the hypergeometric series takes algebraic values. In

order to do this, they first determinate in each case an infinite subset on which

the hypergeometric series may take algebraic values (the key here is Wüstholz's

theorem). Secondly and thanks to an idea of Beukers, they prove an identity relat¬

ing the hypergeometric series with some modular functions. This identity for the

Introduction 3

hypergeometric series enables Beukers and Wolfart to show that the series indeed

takes an algebraic value at each point of the selected subset. In each case, theycalculate one algebraic value of the hypergeometric series. Using the identity for

F( j^, ^, \\ ^rO, Flach [F189] calculated two more algebraic values of this series.

Using only classical transformations for hypergeometric series and some known

values of the complete elliptic integral K(k) = \jiF{\^, \, \\k2), Joyce and Zucker

[JZ91] calculated two new algebraic values, one of them correcting one of Flach's

ones. They also expressed four more values of this series in terms of values of the

Gamma-function.

In our turn, we have calculated four new algebraic evaluations of F(-^, —, i; z).

For this, we have calculated the value of J (mi) and —^ for some m e N in

using long known results on modular equations (see Weber [We]) and inserted them

into the identity for F(j^, -^, \; ^f^). Moreover, using the same idea as Beukers

and Wolfart, we have produced twenty new identities for fifteen hypergeometricseries. Some of them have allowed us to show the algebraicity of the corresponding

hypergeometric series at the points of some infinite subset of Q.

«eft I I tafci J iXm.1 fan* I

Chapter 1

Abelian Varieties Associated to

Hypergeometric Series

1.1 Definitions and Basic Facts on Hypergeometric

Series

The Gauss' hypergeometric series was first introduced by Euler (1778) as the fol¬

lowing power series in the complex variable z

F(a, b, c; z) = > ——- z,

^ (c;n)(l;w)7Î=U

where a, b, c e C, c/0,-1, —2, ...and

(x;n) :=x(x + 1) •

...

• (x + n — 1) if n > 0

1 if 7i = 0.

If a or b is a nonpositive integer, then the series reduces to a polynom. In any case,

this series defines a holomorphic function of z in the open unit disc. If Re(c — a —

b) > 0, the series also converges on the unit circle ([IKSY] 1.2). When Re{c) >

Re(b) > 0 and |z| < 1, the hypergeometric series has an integral representation

i rlF^'b'c'^ =

^7Z ZT/ xb-la-x)c-b-l(l-zx)-adx, (1.1)

B(b,c-b) Jo

5

6 1. Abelian Varieties Associated to Hypergeometric Series

where B(b, c — b) — fQ xb~l(\ — x)c~b~ldx is the classical Beta-function. It can

be shown that the hypergeometric series satisfies a linear differential equation of

order 2 with three regular singularities for which F(a, b, c; z) is the holomorphicsolution at 0 taking value 1 (more on this aspect in §2.4.2).

The path of integration of the integral in the numerator of (1.1) is the open real

segment (0, 1) in C. The integrant xb~l{\ — x)c~b~l(\ — zx)~adx can be viewed

as a differential 1-form on the (singular) algebraic projective curve C(N, z) defined

by the equation

yN =xA(\-X)B(\-ZXf^

where N is the smallest common denominator of a, b, c and

A:=N(l-b), B:=N(\+b-c), C := Na.

If 1 — b, 1 + b — c, a are not integers (i.e. N \ A, B, C), we can replace the

integral fQ xb~] (1 — x)c~b~l (1 — zx)~adx, up to an algebraic factor, by a period

f ^y on C(N, z), where y is a cycle on C(N, z) whose image under the projection(x, y) \-> x is a double contour loop (or "Pochhammer" loop) around 0 and 1. More

precisely, the relation is

dx T1— = d-Â)(l-^£) / xb-\l-x)c-b-\\-zx)-adx,

y y JO

where £>/ = e~N~. In particular, / y-is defined for z e C — {0, 1} without condi¬

tions on Re(b), Re(c).

In an analogous way, the integral B(b, c — b) appearing in the denominator of (1.1)can be considered, up to an algebraic factor, as a period on a projective algebraiccurve C{M, 0) defined by the equation

yM = xp(l-x)Q,

where M is the smallest common denominator of b and c and

P:=M(l-b), Q:=M(l+b-c).

Because MIA'', C(M, 0) is isomorphic to a component of C(N, 0). (Remark that

even if C(N, z) is irreducible for z ^ 0, 1, C(N, 0) may not be so!)

1.1. Definitions and Basic Facts on Hypergeometric Series

The Gauss' hypergeometric series has been generalized in many ways to series in

several variables. We will however be concerned only by the following Appell-Lauricella function F\ (see [CW91] §1) of the d complex variables x2, ..., xj+y

Fy(a,b2,...,bd+1;c;x2,...,xd+1) = ^(c. T , ^ rr (l.n.^

llxj '

where the parameters a,b2, -, bd+\, c lie in C, c ^ 0, — 1, —2, ...and j runs from

2 to d + 1. This series converges when, Vj = 2, ...,cl + 1, \xj\ < 1. It is also

denoted by Fd if d > 2. If Re(c) > Re{a) > 0, it has the following integral

representation

5(,

-i pood+1

/ u-c+^Jb'(u - \)c-a-x r\(u-Xj)~b>du. (1.2)a,c-a)Ji l i

7=2

If the exponents are not integers, the integral can be replaced up to an algebraic

factor by a period

d+\

JyJ=2

of a differential 1-form on a curve along a cycle on this curve whose projection in

C is a double contour loop. The relations between the parameters are

d+l

/ao = c — ^ bj, jjb\ = 1 + a — c and, for j e {2, ...,d + 1}, fij = bj.

7=2

You will find more on the definition of this curve in § 1.4.1.

Remark 1. Because F {a, b, c; z) is symmetric in a and b, it has also the integral

representation

1 '!

B(a, c — a) Joxa'L(l - x)c-a-L(l - zxyDdx.

If - replaces x in the integral, this becomes

1 C°°/ u-c+b(u - \f-a-l{u - z)~°du,

,c-a) JyB{a

showing agreement with the generalization (1.2)


1.2 Motivation and Structure of the Chapter

As quoted in the Introduction, the goal of this chapter is to construct abelian vari¬

eties for hypergeometric series in several variables. We have seen in Section 1.1 that

the hypergeometric series in several variables has an integral representation which

can be interpreted, under some conditions on the parameters, as a quotient of alge¬braic multiples of periods on algebraic curves, hence on their lacobian variety. If

the parameters and the variables lie in Q, the curves and their Jacobian variety are

defined over Q. Then, according to Wiistholz's result (Satz 2 in [WW85]), such a

quotient

f co?

Jy2 A

of periods f cot on abelian varieties At (with A2 simple) can only be algebraic if

A2 is isogenous to a factor of A\. This critérium gives a powerful tool for testingthe algebraicity of hypergeometric series.

In the case of hypergeometric series in one variable, this has strongly be used byWolfart (in [Wo85], [W088], [BW86]), after he has defined the abelian variety asso¬

ciated to the numerator of the integral representation of the Gauss' hypergeometricseries. The abelian variety associated to the denominator was already known, since

it was constructed by Rohrlich and Gross in [GR78].

In this chapter, we construct precisely the abelian variety on which an algebraic

multiple of the numerator of the integral representation of the hypergeometric series

is a period. This is done in Section 1.4 for Appell-Lauricella hypergeometric series

(i.e. for hypergeometric series in any number of variables). Section 1.3 introduces

the theoretical material necessary for the constructions and the results of Section

1.4. Section 1.5 shows how the abelian varieties for the case of one variable (i.e. for

Gauss' hypergeometric series) can be deduced from our general construction. The

results obtained in this section will respectively prove and correct some assertions

of Wolfart, [W088]. The abelian varieties associated to the denominator (which is

always a Beta-function!) will also appear as special cases of our results. This is

shown in Section 1.6. Finally, in Section 1.7, the definition of the family of curves

treated in Section 1.4 is enlarged by admitting some "degenerated" curves. These

curves can be considered as fibers above (P^)r+1. The genus of the irreducible

fibers and the dimension of the corresponding abelian varieties are calculated.

1.3. Preparatory Material 9

1.3 Preparatory Material

Throughout this section k will be an algebraically closed field.

1.3.1 Affine Algebraic Varieties

Let k be an algebraically closed field and T ç k[x\, ..., xn] a set of polynomials in

n variables over k. The common roots in kn of all polynomials in T build a subset

Z(T) of kn:

Z(r):={Pet";/(P) = 0,V/er}.

Definition 1.1. A subset Y ç kn is called an algebraic set, if there exists a subset

T ç k[x\, ..., xn] suchthat Y = Z(7).

The intersection of any family of algebraic sets is an algebraic set and the union of

a finite number of algebraic sets is again algebraic, hence we can define a topologyon kn by defining the closed subsets to be the algebraic sets. (Note that 0 = Z(l)and kn = Z(0)). This topology on kn is called the Zariski topology.

Definition 1.2. The set kn equipped with the Zariski topology will be called the

affine n-space over k and denoted by A£ (or simply A").

Definition 1.3. An affine algebraic variety (or affine variety) over k is a nonemptyclosed subset of A^ (together with the induced topology).

The set of all polynomials in k[x\, ..., xn] which vanish at all points of an affine

algebraic variety içAJ constitue an ideal of k[x\, ..., xn]:

J(X) := {/ e k[xu ..., xn]; f{P) = 0, VP e X}.

According to Hilbert's so-called Nullstellensatz, which implies that any polynomial

g in k[x\, ..., xn] which vanishes at all points of Z(J{X)) has a power which be¬

longs to J (X), one can associate a unique ideal I (X) to X by demanding it to be

radical (that means I(X) = */I(X), where s/I(X) = {f Gk[x\, ...,xn]; 3r >0 s.t.

fr e I(X)}).

Remark2. I(X) is finitely generated, because the ring k\x\, ..., xn]isnoetherian. In

particular, each affine variety is the locus of zeros of a finite number of polynomials.

Definition 1.4. If X ç A£ is an affine variety, the affine coordinate ring of X is

defined to be the quotient ring k[x\, ..., xn]/I (X). It is denoted by k[X].


1.3.2 Projective Algebraic Varieties

The procedure to define a projective algebraic variety is very similar, the main dif¬

ference is that we work in the projective space ¥n(k) (or JcW1 or simply P"), what

forces us to use homogenous polynomials.

First recall that Fn(k) is the set of equivalence classes of nonzero in + 1)-tuples(flo, ••> <3«), oil k, under the equivalence relation

(flo, •-, on) ~ (a'Q, ..., a'n) -O- 3À e k* s.t. al = ka[, V; e {0, ...«}.

The class of (oq, ..., an) is denoted by (ao :...

: an) and is called a point in ¥n(k),while by a set of homogenous coordinates for this point, we mean any représentantof its class.

Let S := k[xQ, ...xn] and Sj be the set of all linear combinations of monomials

in xq, ..., xn of total degree d, then Sci ç. S with the property that for any e,f>

0, Se Sf ç Se+f. This gives the ring S the structure of a graded ring. An element

of Sd is called a homogenous polynomial of degree d.

Anyone polynomial in S would not define a function on Vn(k), because it has to

take the same value on all représentants of the same equivalence class in (kn+1 —

{0})/~ = P"(fc).

However, the vanishing of a homogenous polynomial is well-defined and, for any

set T of homogenous polynomials in S, we denote the set of their common zeros by

Z(T) := {P e F\k); f(P) = 0,Vfe T},

and define

Definition 1.5. A subset Y of Pn(&) is an algebraic set, if there exists a set T of

polynomials in k[xo, ..., xn] such that Y = Z(T).

The algebraic sets in Pn (k) satisfy the properties requiered for them to be the closed

sets of a topology on Tn(k). This topology on Fn(k) is called the Zariski topology.

Definition 1.6. ¥n(k) together with the Zariski topology is called the projective

n-space.


Definition 1.7. A projective algebraic variety (or projective variety) over k is a

nonempty closed subset of P (k) together with the induced topology.

Let 7 be a projective algebraic variety in P" (k) and define the homogenous ideal

I(Y) of F to be the ideal in k[xo, ...,xn] generated by

{/ G k[%Q, ..., x„]; / homogenous and f(P) = 0, VP e Y}.

As in the affine case, Y is actually the locus of a finite number of homogenous

polynomials.

Definition 1.8. The ring k[xo, ..., xn]/I(Y) is called the homogenous coordinate

ring of Y. It is denoted by k[Y].

Remark 3. P" (Jc) is covered by a finite number of open subsets, each of them home-

omorphic to A^ and, consenquently, every projective variety admits a finite covering

by open sets homeomorphic to affine varieties.

Terminology 1. The qualifier algebraic will be used for a variety to mean that it is

either affine algebraic or projective algebraic.

1.3.3 Irreducibility and Dimension

Definition 1.9. A nonempty subset in a topological space X is irreducible if X =

Xi U X2 with X\ and X2 closed subsets of X implies X\ = X or X2 = X. A

nonempty subset of X will be called reducible if it is not irreducible.

Definition 1.10. Let X be a topological space. The dimension dim(X) of X is

defined to be the supremum of all integers n such that there exists a chain Zq c

Zi C ... C Z„ of distinct irreducible closed subsets of X.

In a noetherian topological space X (that is X satisfies the descending chain condi¬

tion i.e. for any sequence Y\ 2 Yi 5 ••of closed subsets, there exists an integer r

such that Yr = Yr+\ = ...), every nonempty closed subset Y can be expressed as a

finite union Y\ U ...U Yn of irreducible closed subsets Yx. If we require that Yt ^ Yj

for i ^ j, then the Yi's are uniquely determined and are called the irreducible

components of Y.


Definition 1.11. The reducibility nature of an algebraic variety is that as a topolog¬ical space.

It can be shown that A£ and Tn(k) are noetherian topological spaces. Thus, each

affine (resp. projective) algebraic variety can be expressed uniquely as a finite union

of irreducible affine (resp. projective) varieties, no one containing another.

Definition 1.12. The dimension of an irreducible affine (resp. projective) algebraic

variety is its dimension as a topologal space. The dimension of a reducible affine

(resp. projective) algebraic variety will be the maximum of the dimensions of its

irreducible components. ([Shi] 1.6.1)

1.3.4 Algebraic Curves

Definition 1.13. An affine (resp. projective) algebraic curve over k is an affine

(resp. projective) algebraic variety over k of dimension 1. An affine (resp. projec¬

tive) algebraic plane curve is an algebraic curve in A2, (resp. P2(&)).

An affine (resp. projective) algebraic plane curve is then the locus of zeros of one

nonconstant polynomial mk[x,y] (resp. one nonconstant homogenous polynomialin k[xç,, x\, X2J). Moreover, it is irreducible if and only if the polynomial is irre¬

ducible and the irreducible components of the curve correspond bijectively to the

irreducible factors of the polynomial ([Shi] II. 1).

Given now an affine plane curve C defined by the equation f(x,y) = 0, where

/ e k[x, y], we would like to show how we can associate to C a projective planecurve and what this means geometrically. First observe that A2, embeds to P2(£) by(x, y) i-> (1 : x : y). Write x := g, y := g and define fhom e k[x0, xi,x2] by

A xl x2fhomfro, XI, x2) := *o/(—> —).

XQ Xq

where d is the degree of /. One can show that the projective plane curve Z{fhom) ^

¥2(k) is the disjointe union of C = Z(f) ç A2, (points with coordinate xq ^ 0)with a finite number of points lying at infinity (points with coordinate xo = 0) and

is actually its closure with respect to the Zariski topology.


1.3.5 Morphisms and Birational Equivalence

In this paragraph, we introduce different kind of maps that exist between algebraicvarieties.

Definition 1.14. Let Y be an affine (resp. projective) algebraic variety over k and

P e Y. A function / : Y -> A^ is said to be regular at P if there is an open

neighbourhood U of P and polynomials g,h e k[x\, ...xn] (resp. homogenous

polynomials g, h e k[xQ, ..., xn] of the same degree) such that h does not take the

value zero on U and / = | on U.

Let W be an open subset of Y, a function Y —> A], is called regular (on W) if it is

regular at every point of W. The set of all regular functions on W is a ring, which

will be denoted by 0(W) (or <DY(W)).

Remark 4. A regular function is continous (for the Zariski topology). It follows that

two regular functions on an algebraic variety Y which agree on a dense open subset

of Y are equal.

Remark 5. ([Shi] 1.2.2) If X is an affine variety over k, then the coordinate ring of

X and the ring of regular functions on X are isomorphic via the map

k[X] — 0(X)

[p] i—> p\x-

It is also interesting to consider functions which are not regular on the whole of X,

but almost everywhere, that is on a dense subset of X. In view of Remark 4, we will

have to identify functions which agree on a dense subset of X.

Definition 1.15. Let X be an algebraic variety over k. We define an equivalencerelation on the set of pairs (U, f), where U is an open dense subset of X and / a

regular function on U, in the following way:

(U, f) ~ (V, g) & flunv = glunv-

The equivalence classes for this relation are called rational functions on X.

Let's now define maps between algebraic varieties which preserve the structure, or,

weaker, which preserve it almost everywhere.


Definition 1.16. Let X and Y be two algebraic varieties. A morphism cp : X —>- Y

is a continous map such that for every open subset V ç Y and for every regularfunction / : V -> A^, the function / o cp : ip"1 (7) -> A], is regular.

Definition 1.17. A morphism is an isomorphism if it admits an inverse which is

again a morphism.

Definition 1.18. Let X and Y be two algebraic varieties. A rational map is an

equivalence class of pairs (U, <p), where U is an open dense subset of X and cp a

morphism U —> Y, under the equivalence relation:

(U, <p) ~ (V, f) <£> <p\unv = f\unv-

A rational map will often be denoted by a représentant of its equivalence class.

Rational maps allow us to define an equivalence relation between algebraic vari¬

eties, which provides a tool for the classification of algebraic varieties.

Definition 1.19. A birational map cp : X -> Y is a rational map which admits an

inverse. By an inverse, we mean a rational map \fr : Y —> X such that i/r o <p =

idx and <p o \j/ = idy as rational maps (i.e. the compositions are regarded on the

intersections of the domains of definition). Two algebraic varieties X and Y are

said to be birationally equivalent (or birational), if there exists a birational map

between them.

1.3.6 Blow-up

As we have seen in Remark 3, §1.3.2, a projective variety is locally homeomorphicto an affine variety. Because the blow-up is a local construction, we do not loose

generality if we restrict ourselves to the blow-up of a point in A£. Up to a linear

change of coordinate, we can even suppose this point to lie at 0. For our purpose,

it is sufficient to explain the construction in the case n = 2 (for general n, see [H]

1.4).

This construction furnishes an example of a rational map which may not be bira-

tional. It also provides a better understanding of rational maps in general, but this

is not the route that we will explore (for more information, consult [Ha] Thm 7.21).Instead, we will use it in resolving singularities on a plane curve (see §1.3.7). More

generally, the blow-up is the key tool in resolving singularities on varieties.


First consider the product A2 x P1, which is an open dense subset of the projectivevariety P2 x P1, and the closed subset

X = {(xi, x2; yi : y2) e A2 x P1; xxy2 = x2y\}

of A2 x P1. The projection A2 x P1 —> A2 onto the first factor induces a morphism

 A2.

Definition 1.20. The set X together with the morphism A2 is called the

blow-up of A2 at the point (0, 0).

A nonzero point in A2 has exactely one preimage under <p, while the point (0,0) e

A2 (which will be denoted by 0) has infinitely many preimages which are in bi-

jection with the lines in A2 going through 0. One can show that cp induces an

isomorphism of varieties from X — cp~l {0} to A2 — {0} and that X — cp"1 {0} is dense

mX.

Also useful in the practise is the description of cp in the local coordinates of X. For

i = 1,2, let (Ui, iii) be the chart of X given by Ut = {(xi, x2; y\ : y2) e X; yt ^ 0}

(note that X = U\ U U2) and i/^ {{x\ ,x2;y\ : y2)) = (-7-, x{), where j is the element

of {1, 2} different from i. Write (ut, vt) for the coordinates of the chart Ulti = 1,2.

Then cp is locally given by

cp o i/TjT (u\, v\) = (v\, u\v\) and

<p o tA2"1(m2, v2) = (u2v2, v2).

Definition 1.21. Let 7 be a closed subvariety of A2 containing the point 0. Let's

define the blow-up Y' of Y at the point 0 to be the closure of cp"1 (Y — {0}), where

(p : X —> A2 is the blow-up of A2 at 0, as described above. The restriction of cp to

Y' is sometimes again denoted by cp.

Y' is a closed subvariety of X and the morphism (p\y induces an isomorphism fromY' — </?_1{0} to Y — {0}, so that Y and Y' are birationally equivalent.

Definition 1.22. The blow-up Y' of F at 0 is also called the proper (or strict)transform of Y in the blow-up of A2 at 0, while the inverse image </?_1{0} c X is

called the exceptional divisor of the blow-up.

An example of blowing-up will be given at the end of the next paragraph.


1.3.7 Desingularization of a Plane Curve

It is now the last possible stage to define singular points. For commodity reasons

and because only such singularities will here be met, we will only define singular

points on plane curves.

Definition 1.23. Let C be an affine (resp. projective) plane curve given by the

polynomial equation f(x,y) = 0 (resp. F(xq, x\, xi) = 0). A singular point (or

singularity) on C is a point on C whose coordinates anihilate the two (resp. three)

partial derivatives of / (resp. F).

A point on C which is not singular is called nonsingular or regular as well as a

curve which has no singular point.

The following notion gives a measure of "how singular" is a point or rather how

serious is the singularity.

Definition 1.24. The multiplicity vp (C) of a point P on C is the order of the first

nonvanishing term in the Taylor développement of / (resp. F) at P.

Note that one has

vp (C) = 1 <£=> P is regular.

Let's now explain how one can resolve singularities on an irreducible affine planecurve C by blowing-up ([Shi] IV.4.1). Let P be any singular point on C and blow

it up. For the preimages of P under the blow-up C, one has

vp(C) > J2 VQ(C'1-Qe<p~HP}

If P is regular, it has exactely one preimage. Suppose then that P is singular, then,if it has more than one preimage, the multiplicities of these are smaller than that of

P. Blow then up the <2's that are again singular on C'. Continue to blow up each

time you get a singular preimage. It can be shown that, for each singular preimage,

you will once get more than one preimage and so on inductively. Thus, the process

must stop after a finite number of blowing-ups, because the multiplicity decreases

but remains positive by definition. Applying the same to the eventual other singularpoints of C (which are in finite number), we get the following theorem.


Theorem 1.1. ([H] V.3.8) Let C be an irreducible curve in a surface X [i.e. an al¬

gebraic variety of dimension 2, e.g. A^ or ¥2(k)]. Then there exists afinite sequence

of blow-ups (ofsuitable points)

Xn -> Xn-\ ->...

-> X\ -> Xq = X

such that the strict transform C of C on Xn is nonsinguiar.

The same is also true if C is reducible ([H] V.3.8.1 and 3.9 or [BK] §8.4) and more

generally, every curve is birationally equivalent to a nonsinguiar projective curve

([H] 1.6.11).

Definition 1.25. In the context of the theorem, but with C not necessarily irre¬

ducible, write tc for the restriction to C of the composition map and call C the

desingularization of C and it : C -> C the desingularization map. (it is actuallya morphism, because it is the composition of morphisms.)

The desingularization C of a plane curve is again a plane curve (because it is bira¬

tionally equivalent to it) and is unique up to isomorphism ([Ful] Ch7 §5). Thoughit is in general not easy to find its equation, it may be in the practise sufficient to

understand its local structure.

Consider the resolution of a singular point P on a plane curve C. According to

the theorem, this is obtained after blowing-up this point and its preimages a finite

number of times. Since the intersection of two irreducible components of C not

contained in each other is a singular point, the proper transforms on C of the irre¬

ducible components of C do not meet. They are actually severed one from another

by blowing-up as soon as their slopes at the center of the blow-up are different ([H]

1.4.9.1). As the blow-up of C at P is defined to be the closure of the inverse imageof C — {P}, it implies that each of the proper transforms of the branches of C car¬

ries exactely one preimage of P (which is the point added by closure). This can be

transposed locally in the following way.

• The inverse images under 7t of the local branches of C at P have no intersec¬

tion point on C.

• They correspond bijectively to the irreducible factors of the polynomial defin¬

ing C locally at P.

• Each of them carries exactely one preimage of P under the desingularization

map jr.


Example 1. Consider the affine plane curve C in Aj^ which is the locus of zeros

of the polynomial f(x, y) = y6 — x4. The sole singular point of C is (0, 0). As

the polynomial / reduces into two irreducible factors y3 — x2 and y3 + x2,the

desingularization C of C will have two branches, no one of them meeting the other

one, and each of them carrying exactely one preimage of (0, 0) (written 0 from now

on). According to Theorem 1.1, a finite number of blow-ups will be sufficient to

desingularize C. For the successive blow-ups, the local coordinates given in §1.3.6

will be used and for the drawings, A^ will be identify with R2.

Figure 1.1 shows how the curve C : y6 — x4 = 0 looks like. Let's blow up C

at the point 0. The strict transform of C is C' : t2 — s4 = 0 (cf Figure 1.2) and

the preimage on C of 0 is 0. It is still singular on C, though it is not on any of

the irreducible components of C'. Blowing up C' at 0, we get a similar situation (cf

Figure 1.3): the preimage of 0 on the strict transform C" : q2 — p2 = 0 of C is 0 and

is singular only because it lies at the intersection of two irreducible components of

C" (each of them being nonsingular). We then have to blow up again at 0. At least

(see Figure 1.4), the two irreducible components of the strict transform C" have no

intersection point (and stay nonsingular), hence C" is nonsingular. Each of these

components carries exactely one preimage of 0, these being (0, 1) and (0, —1).

1.3.8 Riemann Surfaces and Nonsingular AlgebraicCurves over C

We first recall the definition of a Riemann surface and then explain how an irre¬

ducible nonsingular algebraic curve defined over C can be given the structure of a

Riemann surface.

Definition 1.26. ([FK] 1.1) A Riemann surface Mis a connected Hausdorff topo¬

logical space on which there exists a maximal set of charts {(Ua, (pa)}aeA (i.e.

{Ua}aeA is an open cover of M and Va e A, 3Va an open subset of C such that

<Pa ' Ua —>- Va is a homeomorphism) with the property that, if Ua HUß ^ 0, then

the so-called transition function

<pa o (pjx : (fß(Ua n uß) —> (pa(Ua n uß)

is holomorphic. For each a, the elements of Va are called local coordinates or local

parameters of M on Ua and Ua is called the domain of the chart {Ua,(pa).

1.3. Preparatory Material19

6_

-V.4Figure 1.1: y = x

t=s"

/

\t=-£

2_

„4

Figure 1.2: r2~ s

1. Abelian Varieties Associated to Hypergeometric Series

q

\

\//q=p

\

\

\

\

\

\

\q=-p

Figure 1.3: q2 = p2

V

v=l

v=-l

Figure 1.4: v = ±1


Hence a Riemann surface is a connected one-dimensional complex analytic mani¬

fold, or also a connected two-real-dimensional differential manifold on which the

transition functions are not only C1 but even holomorphic.

Classical examples are the complex plane and the Riemann sphere C U {00}. A less

classical one is the desingularisation of an algebraic curve defined over C as we will

see in the second part of this paragraph.

Originally, Riemann had defined such surfaces in order to "unformize" multivalued

complex functions of one variable, such as taking the complex n-th root or the

logarithm of a complex number. Indeed, these functions can be defined as uniform

functions on a finite covering of C.

Definition 1.27. Let TV and M be Riemann surfaces. A continous map f : M -> N

is called holomorphic (or analytic) if for every chart (U, cp) on M and any chart

(V, i/O on N such that U n f~l (V) ^ 0, the map

f o / o cp-1 : <p(U n f-\V)) — f(V)

is holomorphic. A holomorphic map M -> C is called a holomorphic function on

M.

A similar definition is applied to a function W -> C defined on an open subset W

of a Riemann surface by considering the intersections with W of the domains of the

charts of M.

Let's consider now an affine algebraic plane curve X defined over C (i.e. X C

A^,) and write X(C) for the set of complex points of X. If x is a nonsingular

point of X and t\, ti coordinates on A^, then by the Implicite Functions Theorem,

one of the two coordinates (say t^) can be written in the neighbourhood of x as a

holomorphic function of the other. Hence the projection on the first factor defines

a homeomorphism of {{t\,t-i) e X(C); \t,\ < rj,i = 1,2} (where r\ > 0) onto

an open domain of C. This defines a new topology on X(C), which is called the

complex topology. The same can be done for a projective plane curve and, more

generally, for any algebraic variety over C ([Sh2] VII. 1.1). In particular, if the curve

X has only nonsingular points, X(C) is locally homeomorphic to a domain of C.

Let then X be a nonsingular projective algebraic curve defined over C and X(C) the

set of complex points of X with the complex topology defined above. The followingproperties hold.


• X(C) is compact ([Shi] II.2.3).

• X(C) is triangulable (for a definition see §1.3.9) ([Sh2] VII.3.2).

If X is moreover irreducible, then

• X(C) is connected ([Sh2] VII.2.2).

Suppose now that X is irreducible. In order for X (C) to be a Riemann surface, the

transition functions must be holomorphic. To show that they actually are, we use the

fact that if / is a regular function atxel and cp : U —^ V <^Ca homeomorphismfrom an open neighbourhood U of x and an open subset V of C, then the function

f o cp'"1 is analytic in a neighbourhood of cp(x). Conclude by remembering that

a projective curve is locally isomorphic to an affine one and that, in this case, the

maps are regular functions, since they are projections on the first (or second) factor.

(See also [Mir] 1.2 §5 for the affine case and 1.3 §2 for the projective one.)

This also holds in bigger dimension: for any nonsingular algebraic variety X over

C, X(C) has the structure of a complex analytic variety ([Sh2] VIII. 1.1).

Hence, if X is a nonsingular irreducible projective algebraic curve, then the set

X(C) of complex points of X admits a structure of compact Riemann surface.

Remark 6. Let X be a nonsingular projective algebraic curve over C and X(C) the

associated complex analytic surface. If / is a regular function on X, then it is a

holomorphic function on X(C), because it is locally the quotient of a polynomial

by a nonvanishing polynomial. By Serre's GAGA theorem, the converse also holds

(see[Sh2] VIII.3.1).

1.3.9 Topological Invariants of a Riemann Surface

In this paragraph, we shall discuss some topological invariants of Riemann sur¬

faces, such as the Euler characteristic and the first homology group with integercoefficients. These objects can be defined for any topological space, but are easier

to calculate if the space is trangulable. Since any Riemann surface is triangulable,we will define these objects through the use of a triangulation. We naturally begin

by explaining what a trangulation is ([AS] Chi §22).


Definition 1.28. An (abstract) simplicial complex K is a set V{K) together with

a family of finite subsets of V(K), called the simplices, such that the following

properties hold:

1. Each a e V(K) belongs at least to one and at most to a finite number of

simplices.

2. Every subset of a simplex is a simplex.

The dimension of a simplex is one less than the number of its elements. A simplexof dimension q is also called a q-simplex. The dimension of the simplicial complexK is the maximum of the dimensions of its simplices (or infinite, if there is no

maximum).

To each q-simplex a of a simplicial complex K can be associated a closed subset

\a\ of M9+1 in such a way that, Vcri, 02 e K, \a\\ n |<r2| = \<x\ C\ 021. The set

\K\ := [Jaefc \<y\ can be given a topology. Together with this topology, \K\ is

called the geometric realization of K.

Definition 1.29. Let F be a différendable manifold of dimension 2. A triangu¬lation of F is the data of a simplicial complex K of dimension 2 together with a

homeoraorphism of \K\ onto F.

It can be shown that every Riemann surface M admits a trangulation ([Sp] Ch 9-

3) and that M is compact if and only if it admits a finite triangulation (that is a

triangulation whose underlying simplicial complex contains only a finite number of

points ([Sp] Ch5-1)).

We would like now to define the homology of a simplicial complex K with coeffi¬

cients in a ring R. For this, we have to define ordered q-simplices and identify someof them.

An ordered q-simplex is a (q+l)-uplet (vo, ..., vq) e V(K)q+1 suchthat {vo, ..., vq)lies in K. Two ordered q-simplices (vo, ...vq) and {vq , ..., vq') will be equivalentif and only if there exists an even permutation s e Aq+\ such that Vi = 0, ..., q,

vi' = vS(i). A class for this equivalence relation is called an oriented q-simplex of

K. Now, define Cq (K, R) to be the free R-module on the set of oriented q-simplicesof K. Let a be an oriented q-simplex, we want to identify the q-simplex <r# having


the inverse orientation with the the opposite —a of a in Cq(K, R) .Let then I be

the R-submodule of Cq(K, R) generated by {a + cr#; cr oriented q-simplex of K},we set

Cq(K,R):=Cq(K,R)/I.

A class [vq, ..., vq] g Cq(K, R) is called a q-chain.

We have a boundary operator dq : Cq (K, R) -> Cq-$K, R) given by

<?

dq([V0, ..., Vq$ = ^2(-l)l[v0, .... Vit ..., Vq],i=0

where the hat"on vt means that the element vt is omitted. This is a homomorphismof R-modules. The image of a q-chain is a (q-l)-chain called its boundary. One

verifies that 3?_io3fy = 0 (i.e. Imdq C Kerdq-\). Hence we have a chain complex

... -+Cq+i(K, R) ^> Cq(K, R) X <ViCSr, R) -^... ^C0(K, R) X 0,

where 3o denotes the zero homomorphism (cf [AS] Chi 23C).

This sequence is not necessarily exact and its nonexactness is measured by the R-

modules

Hq(K, R) := Ker(dq)/Im(dq+i) where q N.

Definition 1.30. For q e N, the R-module Hq (K, R) is called the q-th (or q-

dimensional) homology R-module of the simplicial complex K. The elements

of Ker{dq) are called q-cycles and those of Im(dq+\) q-boundaries.

For a triangulable surface (or manifold) F, we define its homology to be that of

its trangulation. It can be shown that these R-modules are actually isomorphicto given R-modules depending directely on the topology of F (these are the so-

called singular homology R-modules) and hence indépendant of the choice of the

triangulation. This justifies the following definition.

Definition 1.31. Let M be a Riemann surface and R a ring. Let the simplicialcomplex K furnish a triangulation of M, then we define the q-th (or q-dimensional)simplicial homology R-module Hq(M, R) to be Hq(K, R).


Remark 7. Since a Riemann surface M can be trangulable by a simplicial complexof dimension 2, we have,

V^>3, Hq(M,R)=0.

From now on suppose that R = Z ([AS] Chi 25). What can be said about Ht (M, 2?)when/ e {0, 1,2}?

• Note that any nonzero multiple of a 0-simplex (vertex) «o is nonzero in

Hq(M, R). On the other hand, take any other vertex a\. In view of the con¬

nectedness of M, there exists a finite sequence of 1-simplices which begins at

a.Q and ends at u\. Hence, a\~

«o and Hq{M, R) = R{œq} — R.

• It can be shown that H2(M, Z) is a free Z-module of rank 1 if and only if M

is compact. In all other cases, H2(M, Z) = 0.

• For a compact Riemann surface M, H\ (M, Z) is a free Z-module of finite

rank.

Definition 1.32. Let M be a compact Riemann surface. For i e {0, 1, 2}, the (finite)rank of Ht (M, Z) over Z is called the i-th Betti number and denoted by b{.

The Betti numbers of M are topological invariants of M (that means that they de¬

pend only on the topology of M and are invariants under any homoemorphism).

Definition 1.33. ([Sp] 5-9) Let K be a simplicial complex giving a triangulationof the compact Riemann surface M. Write nq for the number of q-simplices in K.

Then, we define

X(M) :- no -n\ +n2.

This number is called the Euler characteristic of M.

This definition is actually indépendant of the choice of the triangulation of M. In¬

deed, the Euler-Poincaré formula

X (Af) = bo~h+b2

shows that x (M) depends only on the topology of M and is therefore a topologicalinvariant of M (see [Sp] 5-9).


For a compact Riemann surface M, we have seen that &o = £2 = 1- In this case, it

follows from the formula that

fci=2-X(M). (1.3)

Next, we will explain how the Euler characteristic of a Riemann surface can be

deduced from that of another compact Riemann surface if there exists a nonconstant

holomorphic map from one to the other. This is the object of Hurwitz formula.

We first begin with a somehow more general definition.

Definition 1.34. ([FK] 1.2.4) Let M and M* be two manifolds. Then M* is said to

be a (ramified) covering manifold of M if there exists a map / : M* —> M (called

covering map) such that for each point P* e M*, there exists a local coordinate

z* around P*, which vanishes at P*, and a local coordinate z around f(P*), which

vanishes at f(P*), and an integer n > 0 such that / is given in terms of these

coordinates by

z = (z*y.

The integer n depends only on the point P*. If n > 1, then P* is said to be a

ramification point of order n. n is called the ramification index (or ramification

order) of P* and is denoted by rf(P*). If n = 1, P* is said to be regular and

we set rf{P*) — 1. If / has no ramification point, then the covering is called

unramified (or smooth).

Remark 8. ([Sp] 4-1) The expression z* i-> (z*)n of / in local coordinates shows

that the ramification points are isolated. Indeed, the chart at P* can be chosen such

that the restriction of the map z* i-> (z*)n to any open subset in the domain U of the

chart which does not contain 0 is in bijection with its image (restriction to a branch

of the n-th root) and consequently each point in U distinct from 0 parametrizes a

regular point on M*.

If M* is compact, the above implies that the covering map / has only finitely manyramification points.

Returning to the case of Riemann surfaces, suppose that M and iV are compactRiemann surfaces and / : M -> N a holomorphic map, then / is either constant

or surjective. If / is nonconstant, it can be shown that at each point of M, there

exist local coordinate as in the definition. Moreover, such a map takes each values


the same number of times counting multiplicities ([FK] 1.1.6), i.e ]meN such that

Vg e N, we have

m = 2_, rf(pyPef-HlQ})

The nonnegative integer m is called the degree of the map / and / a m-sheeted

cover of N by M.

Remark 9. A point Q e N has exactely m distinct preimages if and only if VP <s

f~l ({Q}), rf(P) — 1> that is all the preimages of Q are regular points of /.

Let's summarize all of this in a proposition.

Proposition 1.2. Let M, N be two compact Riemann surfaces, then any noncon-

stant holomorphic map f : M -> N is a covering map and its degree equals the

number ofpreimages ofeach point ofN, exceptfor a finite number of isolated ones.

If there exists a map as in the proposition, then the Euler characteristics of M and

N respectivelly are related in the following way.

Proposition 1.3. HurwitzformulaLet M and N be two compactRiemann surfaces and suppose that there is a noncon-

stant holomorphic map f : M —>- N. Write m for the degree of f and P\, ..., Ps e

Mfor its ramification points. Then one has

s

x(M)=m-X{N)-YJ(rf{Pi)-\)-i=\

Proof ([FK] 1.2.7) Let K be a simplicial complex furnishing a triangulation of

N. Since the ramification points of / are isolated, so are their images and we can

suppose these to coincide with O-simplices of K if only to refine the triangulation.(Recall that the Euler characteristic is indépendant of the choice of the triangula¬tion.) For ç e {0, 1, 2}, write nq for the number of q-simplices in this triangulationand lift it to a triangulation of M via the map /. This triangulation of M has

m • «2 2-simplices,m -n\ 1-simplices,

m no — Y2i=i(rf(Pi) ~ 1) O-simplices.


Conclude by summing

s

X(M) = m(n0 - «i + n2) - J](r/(P(0 - 1).

i=l

D

1.3.10 Tangent Space to an Algebraic Variety and Differential

Maps

We first give a geometric definition of the tangent space to an affine algebraic variety

(cf [Shi] II. 1.2) and then relate it to an algebraic object, what will justify the generaldefinition.

Let X c A£ be an affine algebraic variety over k. According to §1.3.1, it is the

locus of a finite number of polynomials in k[x\, ..., xn]. Let F\, ..., Fm be these

polynomials and fix a point P on X. Up to a linear change of coordinates, we can

suppose this point to lie at 0. A line L in A£ going through 0 is a set {ta; t e k},where a e AJ is a fixed nonzero point. The intersection of X with L is given by the

equations

Fi(ta) = ...= Fm(ta) = 0.

a being fixed, the equations are polynomial in the variable t and have t = 0 as

solution. Write ft(t) := Fi (ta) fori e {1, ..., m}.

Definition 1.35. The intersection multiplicity of the line L with X at 0 is the

multiplicity of t = 0 as a commun root of all ft, i — 1,..., m. It is indépendantof the choice of generators of I(X) and nonnegative, since 0 g L (1 X. If Vi e

{1, ..., m}, fi = 0, the multiplicity is setted to be infinite.

We can now define the tangency to X of a line going through 0 and the tangent

space to X at 0.

Definition 1.36. A line L is said to be tangent to X at 0 if its intersection multi¬

plicity with X is > 1.

Definition 1.37. The geometric locus of all points lying on lines tangent to X at 0

is called the tangent space to X at 0 and is denoted by ®x,o or simply ©q.


Since 0 is a root of each Fi, i = 1, ...,m (because 0 e X), the Fi's have constant

term equal to 0. Thus, for every i, we can write

Fi = Li + Hi

where L(- is the linear term of Ft and deg{Hl) > 2.

Now, because Vz = 1, ..., m, Lj(ta) = tL{(a), we have

/i(0 = *£,(a) + flKfa), Vi = 1, ..., m,

and the order of 0 as a commun root of a the polynomials fi, i = 1, ..., m, is > 2

if and only if Lz (a) = 0, Vz = 1, ...,m. Hence, the tangent space to X at 0 is given

by a system of linear equations

L1(a) = ...= Lm(fl)=0 (1.4)

and therefore is a linear subspace of A£.

The tangent space to X at any point P of X can be defined in an analogous way. It

is also a vector space over k and will be denoted &x,p or simply &p.

Let X still be an affine algebraic variety in Aj£ and g be a regular map on X, i.e.

g £ Ö(X). Using the isomorphism in Remark 5, §1.3.5, we can identify g with

the restriction to I of a polynomial G in k[x\, ..., xn]. For such a polynomial and

a point P on X with coordinates (p\,..., pn), we define dp G to be the linear form

on the ^-vector space @x,p defined by the the term of first order in the Taylor

development of G at P. That is, for x = (x\,..., xn) g ®x,p, we have

"

?iC

dPG(x):=T—(P)(xl-pi).*-^

OXi1=1

Note that for F, H e k[x\, ..., xn], we have

dp{F + H) = dPF + dPH and

dP{F H) = H{P)dPF + F(P)dPH.

Write again F\,..., Fm for the generators of the radical ideal I(X) of X, then, in

view of (1.4), the tangent space to X at P is given by the equations

dpFi{x) = 0 for i = 1, ..., m, (1.5)

30 1. Abelian Varieties Associated to Eypergeometric Series

because L,- is precisely the linear term of Fj. Now, by Remark 5, we know that

G is unique up to addition by an element of I(X) = (F\, ..., Fm). Let G' :=

G + J2T=\ AiFi with Ai e k[x\, ..., xn], for /' = 1, ..., m, then

m

dpG' = dPG + J2(Fi(P)dpAi + Ai(P)dpFi).

i=i

But Pel implies Ft(P) = 0, Vi = 1,..., m and besides the linear forms dp Ft are

zero on ©x,P by definition of &x,p- Hence

dPG' = dPG

and we can define

Definition 1.38. Let X be an affine variety in A^ and g e Ö(X) represented byG e k[x\, ..., xn], then we define the differential dpg of g at P to be the linear

form dp G on 6p.

As seen above, this definition is indépendant of the représentant of the class of G in

k[X] = k[x\, ..., xn]/I(X) C- 0(X). This definition furnishes a homomorphism of

Ö(X) to the ^-vector space ®P of linear forms on @p. Remark that if a e k, then

dpa = 0, thus for / e O(Z), f(P) e k and we have

dp(f-f(P))=dPf.

It is therefore sufficient to restrict the study to the image of the set of regular func¬

tions vanishing at P. This set TtP = {/ e O(Z); f(P) = 0} is an ideal in O(X)and the quotient ÜJlp/Wlp is a jfc-vector space. The homomorphism dp : Tip —>

®p; f ^ dp f is surjective and its kernel equals Tip. Hence it induces an isomor¬

phism of ^-vector spaces ([Shi] II.1.3.1)

dp : Tip/Til -^ ®p-

The dual spaces are then also isomorphic

®p-(Tip/Til)*. (1-6)

We would like to use this isomorphism in order to define the tangent space to a

projective variety Y. We know (cf Remark 3) that such a variety is covered by a

finite number of affine ones, but the above cannot be applied directely to any affine

neighbourhood of P e Y, because it could depend on the choice of this affine

neighbourhood. This leads us to the definition of a local algebraic invariant of a

point on an algebraic variety.


Definition 1.39. ([Shi] II.1.1) Given an algebraic variety Y and a point P on Y,

consider the set of pairs (U, f), where U is an open neighbourhood of P and / e

0(U), and the following equivalence relation on this set

(U, f) ~ (V, g) ^^ 3W open cUHV s.t. P e W and /|w = g|w.

Write Op (or öy, p) for the set of equivalence classes for this relation. It is a ring for

the operation induced from the usual multiplication and addition on représentantsand has a unique maximal ideal

WlP = {{(U,f))Öp;f(P) = 0}.

Hence Op is a local ring which is called the local ring of P.

When Y is an affine variety, we have

TlP/Tlp ~ Tip/Til (1.7)

and remembering the isomorphism (1.6), we define

Definition 1.40. ([Shi] II. 1.3) Let Y be an affine or projective variety and P e Y,

the tangent space to Y at P will be (Tip/Tip)*, where Tip is the maximal ideal

of the local ring Op. It will be denoted by ©p (or @y,p).

By (1.6) and (1.7), ®y,p is then also the tangent space in the previous sense to any

affine neighbourhood of P.

Now, if Y is any algebraic (i.e. affine or projective) variety and / G 0(7), how can

we associate to / an element of the dual space to ®y,P^ In the affine case, for a

polynomial G e k[x\, ..., xn] and a point P = (p\, ..., pn), we had define dpG to

be the term of first order in the Taylor development

G = G(P) + J] —(P)(xi - Pi) + H,

i-idXi

of G at P. Here H is of degree > 2 and H(P) = 0, hence H e Tip. This implies

G - G(P) = J] —(PXxi - Pi) (mod Tl2P)i-l

dXi


and, by definition,

dPG = G - G(P) {mod VJtp).

This allow us to extend Definition 1.38 as follow.

Definition 1.41. Let Y be an affine or projective algebraic variety, P e Y and

/ e 0(7) (or even in 0P(7)), we define the differential dPf of / at P to be the

following element of (®y,p)*

f-f(P)(modm2P).

Note that for f, g G 0(7) and P e Y, we have

dp(f + g) = dpf + dPg and

dp(f-g) = f(P)dpg + g(P)dPf.

Let's now define the differential of a morphism between algebraic varieties ([Shi]II. 1.3). Let X, Y be two algebraic varieties and $:I^>Fa morphism. If / is a

regular function on Y, then the composition / o <£> is regular on X, hence <t> induces

a linear map <ï>* : 0(7) -> G(X) by sending / to / o c£>. Obviously, for each

P eX, $*(SDT$(P)) C Tip and ^(STt^) C Wtp. Hence l>* factorizes and, for

each P e X, induces

$* : aK*(/>)/9j%(P) — ajip/mij.

To a linear function g : DJlp/Wl2p -> k, we associate g o $* g (9Jt$(p)/9D?|(p))* ~®f,$(P) and define

Definition 1.42. Let X, 7 be algebraic varieties, P e X and O : X -> 7 a mor¬

phism of algebraic varieties, then the induced linear map on the tangent spaces

®x,p —> ®Y,Q{P)g i—> go®*

is called the differential of <£> at P and denoted by dpO.

Note that if <E> is the identity on X, then for every P e X, dp<& is the identity on

®x,p- If Z is another algebraic variety and ^ : 7 -> Z a morphism, then the

differential dp(ß> o $) : ®x,p -» @z,(*o*)(P) is given by

Jp(^/ O O) = d$(p)*I> o JpO.


In particular, if $ : X -> Y is an isomorphism of varieties, then its differential

dpQ? : ®x,P -> ®y,$(P) is an isomorphism of ^-vector spaces.

Remark 10. ([Mil] §4 p.63) A morphism Y, where X ç Am and Y ç A"

are affine algebraic varieties, is given in each coordinate by a regular function. Write

cpi for the regular function in the t-th coordinate of cp and F, for the projection

Cyi, ••. v„) i-> Vj. Then the differential dpcp of cp at P e X is the map ©x,p ->

®y,<p(.P) such that

/I!~

Eo(Pi—(P)dpXj,7=1

J

where Xf is the projection (x\, ...,xm) \-> xt and dpXi : TpX ^ k the linear map

sending QM t0 xi ~ Pi-

1.3.11 Differential Forms on Algebraic Varieties

Given an algebraic variety X over k, we have shown in Paragraph 1.3.10 how can

be associated to each point P of X an element of ©^ p.This was done trough the

use of any regular function f onX; indeed, for each P e X, dpfe ©*x p.For an

open set V of X, the set <J>[V] of functions sending each point P of V to an element

of ®*x pis an abelian group for the usual addition of functions and a 0(y)-module

for the action (/ <p)(P) := f(P)cp(P), where / e 0(V).

Definition 1.43. An element of <f?[V] will be called a differential form (or differ¬

ential 1-form) on V.

The set <3>[X] would actually be too big to be interesting, therefore, we would like

to distinguish special elements of <Î>[X], whose variation on P is better controled,

just as the regular functions on X are better understood that any one function from

X to A|. Among the elements of <&[X], we distinguish those that are, in substance,

locally given by differentials of regular functions on X. Indeed, let's set

Definition 1.44. ([Shi] III.5.1) Let X be an algebraic variety and V an open subset

of X. Then a function <p sending each P e V to an element of &*x pis said to be

a regular differential form (or regular differential 1-form) on V, if each p e V

has a neighbourhood U C V such that the restriction of cp to U belongs to the

34 1. Abelian Varieties Associated to Hypergeometric

Series

Ö(U)-submodule of <ï>[U] generated by the elements df with / e 0(U), where dfis the function on U such that, for Q e U,df(Q) = dç)f.

The set of all regular differential forms on V is a module over G(V), and, in partic¬ular, a ^-vector space, which will be denoted by Œ1 [V] (or ß^[ V]).

If co e Q,l[X], then for each P in X there exist a neighbourhood U of P and

/l. -, /«. Si. -. gm e 0(f/) suchthat

m

Vßef/, (ù{Q) = Y,fi(Q)dQgi-

If X is a nonsingular projective algebraic curve, one can show that the k- vector

space ^[X] has finite dimension ([Shi] III.6.3).

Definition 1.45. Let X be a nonsingular projective algebraic curve. Then the num¬

ber dim^ß1 [X]) is called the genus of X and denoted by g[X] or simply g if there

is no ambiguity.

Remark 11. If X is irreducible and defined over C, one can show ([Sh2] VII.3.3)that

X(X(C))=2-2g[Z], (1.8)

where / (X (C) ) is the Euler characteristic of the Riemann surface X (C) (see § § 1.3.8

and 1.3.9 for the definitions).

Remark 12. ([Shi] III.5.1) If C is an algebraic plane curve over k given locally bythe affine equation f{x, y) = 0 and (xo, yo) a nonsingular point on X, then at least

one of the two derivatives of / does not vanish at (xo, yo)- Without restricting the

generality, suppose that §Hx0, yo) ^ 0. Then by the Implicite Functions Theorem,

y can be written in a neighbourhood U of (xo, yo) as a holomorphic (hence regular)function of x and, because the differentials of the local parameters at (xo, yo) form a

basis of Op for every P at which all parameters are regular, any regular differential

form on C can be written in U as

fax,

where / is a regular function on U.


In §1.3.10, we have seen that a morphism <t> : X —> Y between algebraic varieties

induces for each P G X a linear map dp<ï> : &x,p -> ®y,<P(P)- Hence, if co is a

differential form on Y, then for each point Q g Y, cù(Q) g ©y n,what implies

co(<P(P)) o dp<& G 6| p.This defines a differential form on X, which is called

the pull-back of œ (with respect to <3>) and denoted by $*«. According to the

definition, we have

VP g X, ($*ö)(i3) = (ü($(?))o^$ i.e.

$*<£> = (ùJ O 4>) o c/$,

where JO is the function on X such that for P g X, c/<$ (P) = dp O. If * : Y -> Z

is another morphism of algebraic varieties and rj a differential form on Z, then the

following holds

VP G X, (0*(vl/*?7))(P) = (01/ o <D)*77)(P).

Indeed, both sides are equal to

T}((V o ®)(P)) o dPÇi! o $).

Suppose now that &> is a regular differential form on Y. An interessant question to

ask is whether the pull-back of co under a morphism <3> : X —> Y is again regular.To answer this question, we will work locally. Indeed, choose a point S e X and

an open neighbourhood V of $>(S) in 7. Then there exist f\, ..., fm, g\, ..., gm e

Oy(V) suchthat

m

VßG V, o)(ô) = J>(ß)Jßg,-,

so that, VP G 0_1(V),

($*w)(P) = (û(<b(P))odP®

= Y!1Ufi^{P))d^{P)glodp^

= EZiifioQKndpigio®).

Now, because $ is a morphism and of the regularity on V of the f 's and gi 's, the

fi o O's and g; o $'s are regular on O-1 (V) and <&*co is a regular differential form

In this way, we have shown that the pull-back of a regular differential form under a

morphism is again a regular differential form.


1.3.12 Differential Forms on Riemann Surfaces

As in the case of algebraic varieties, we first have to define the tangent space to a

Riemann surface.

Definition 1.46. ([Sp] 5-4) Let M be a Riemann surface. A (differentiable) pathon M is a pair (c, Ic), where Ic is an open interval in R and c a continous map from

Ic to M which satisfies

VP 6 M such that 3t0 e lc with c(t0) = P and for each chart (U, <p)of M with P G U, the composition cp o c is a differentiable function on

{t e Ic\ c(t) <s £/} and (<p o c)'(to) =£ 0.

A pair (c, 7C), where 7C = (a, è) is an open interval on M and c a continous map

from 7C to M, is called a piecewise differentiable path on M if there exists a

partition a = to < t\ <...

< tr = b of Ic such that, for each i in {1, ..., r}, the

restriction of c to the open subinterval Icl := (^_i, ^) furnishes a differentiable

path(c|/c,,/c>ï-)onAf.

Fix a point P on M. On the set of paths on M going through P (it can be supposedthat 0e/c and c(0) = P), we would like to identify the differentiable paths havingthe same tangent direction at P, Hence, we set

(c\, IC] ) ~ (c2, 7C2) <^=^ 3 a chart ((7, <p) of M such that P e U and

3j(p°ci)[r=o = £(ç0oc2)|,=o-In view of this equivalence relation, we define

Definition 1.47. Let M be a Riemann surface ans P e M. The tangent space

TpM to M at P will be the set of equivalence classes of differentiable paths on M

going through P under the above equivalence relation.

Remark 13. If V is the topological space C", then VP e V, TpV is in bisec¬tion with Cn via the well-defined map ap : TpV —>• C" given by ((c, 7C)) h->

Definition 1.48. Let M, N be two Riemann surfaces, P e M and $ : M —^ AT a

holomorphic map. Then we define the tangential map of <t> at P to be the map

7>0 : TPM —> T${P)N

((c,Ic)} i—> ($oe,/c).


Remark 14. If <$> : M —> N and ty : N -» L are holomorphic maps between

Riemann surfaces, then for P e M, Tp(^ o <J>) = r$(p)^ oTp$ and the tangential

map at P e M of the identity map on M is the identity map on TpM. Hence, if

$ : M -> N is a biholomorphic map, then VP e M, Tp$> : TpM -^ 7$(P)iV is a

bijection.

Using the above definition, we define the differential of a holomorphic function

Definition 1.49. Let / : M —>- C be a holomorphic function on a Riemann surface

M and Vß e C, œq the bijection TgC —>- <C, then we define the differential dp/of / at P <s M to be the composition

otf(P) ° Tpf : TpM —> C.

Remark 15. Let M be a Riemann surface, P e M and (£/, 99) a chart on M such

that P G U. Then y :U —> <p(U) is biholomorphic and

c/p TV(P)C -> C

is a composition of bijections. Hence a C-vector space structure can be defined on

TpM by setting the bijection dpcp to be linear. It can be shown that this structure is

indépendant of the choice of the chart.

The special case N be a holomorphic map between Riemann surfaces

and P e M. Let further (U, <p) be a chart of M at P and (V, \Jr) a chart of N at

O(P). Then the composition

d${P)f o Tp$ o (dP(p)~l : C —> C

equals of(1/fo$)(p) o T<p(P)(f o O o Op^1)) o (o^p))-1 and is given by

3-1

z i— — (iA ° <E> o (<p '»[^(P) • (z - <p(P)).

This being a linear transformation of z, TpO will be linear for the vector space

structures on TpM resp. 7$(P)iV for which dpcp resp. d$(p)i^ is linear.


The differential dp f at P of a holomorphic function /ona Riemann surface M 3

P being the composition of two linear functions it is linear and defines an element

of (TpM)*.

Therefore, as for algebraic varieties, we have here associated to a "good" function

and to a point on the variety a linear form on the tangent space to the variety at this

point. Going further with this analogy, we define

Definition 1.50. Let M be a Riemann surface and V an open subset of M. Then

a function co sending each point P of V to an element of (TpM)* is called a holo¬

morphic differential form (or holomorphic differential 1-form) on V if, for each

chart (U, <p) of M with V n U ^ 0, there exists a holomorphic function / on V (~) U

such that

o\vnu = f d(p.

Then, for each P e V n U, we have

œ(P) = f(P)dp<p.

Remark 17. The set of holomorphic differential forms on an open subset F of a

Riemann surface M is a C-vector space, which will be denoted by ß^n[V]. It can

be shown ([AS] 24A) that, if M is compact, then £^n[M] has finite dimension.

Definition 1.51. Let M be a compact Riemann surface, then the nonnegative inte¬

ger dimc^^JM] is called the genus of M and denoted by g(M).

One can shows that the first Betti number of a compact Riemann surface M actually

equals 2g(M) ([Sp] 5-9) and according to formula (1.3) in §1.3.9, we have

X(M) = 2-2g(M). (1.9)

Comparing with (1.8) in §1.3.11, we deduce that the genus of a nonsingular irre¬

ducible projective algebraic curve defined over C equals the genus of the associated

compact Riemann surface X(C), i.e.

g[X] = g(X(Q). (1.10)

This shows that the vector space of regular differential forms on X and the vector

space of holomorphic differential forms on X(C) have the same dimension. The

following remark makes the comparision even stronger.


Remark 18. Remember Remark 6 at the end of § 1.3.8 which asserts that the regularfunctions on a nonsingular projective algebraic curve X over C are holomorphicfunctions on X(C) and vice versa. This can actually be transposed to differential

forms. Indeed, having regard to Remark 12, §1.3.11, a regular differential form co

on X is locally given on an open subset U of X by felt, where / e 0(U) and t a

local parameter regular on U. Thus / and t are holomorphic on U equipped with

the complex structure induced from that of X (C) and so are their restrictions to the

intersections with the domains of the charts. Hence co is holomorphic on X(C). The

converse also holds (by the same arguments and Remark 6) and we have

Q1[X] = nïan[X(Q].

1.3.13 Integration of Holomorphic Differential 1-Forms on Rie¬

mann Surfaces

In this section, we would like to define the integral of a holomorphic 1-form on a

Riemann surface M along an element of H\ (M, Z). To this aim, there will first be

associated a piecewise differentiable path to a 1-cycle on M.

According to [Sp] 9-3, each Riemann surface can be triangulated in such a way that

the geometric realization of each 1-simplex furnishes, via the homeomorphism to

M, a differentiable path on M. From now on, we will suppose the triangulationto satisfy this. If the 1-simplex is oriented, the paramatrization of the path has to

agree with the orientation. To the sum of two oriented 1-simplices, the endpointof the former being the initial point of the latter, is associated the concatenation of

the corresponding pathes ([Sp] 5-8). Hence to each 1-cycle is associated a closed

piecewise differentiable path on M. (In view of this, the 1-cycle and the associated

path will sometimes be denoted by the same symbol.)

Let (JJ, cp) be a chart on the Riemann surface M and co a holomorphic differential

1-form on M. Then there exists a holomorphic function / on U such that

v\u = fdep.

Let further y be a 1-cycle on M contained in U and suppose that the associated

closed piecewise differentiable path c : / -> U is differentiable. By the way, / can

be supposed to be (0, 1) if only to be reparametrized. In this situation, we define


the integral of co along y to be the complex number

• l

Jy JO

f(c(t))dt(cpoc).

This generalizes to the case where c is piecewise differentiable and c{I) not con¬

tained in the domain of a unique chart. Indeed, there exists a partition 0 = to < t\ <

...< tr = 1 of (0, 1) such that the restriction c; of c to the subinterval Ij := (U-i, tj)

is a differentiable path whose image is contained in the domain U{ of a unique chart

{Ut, (pt) (cf [Mir] IV 3, Lemma 3.7). For i = 1, ..., r, write ft for the holomorphicfunction on Ul satisfying

®\u, = Adept-

Then we set

Consider now a class C in H$M, Z), then two représentants yi, 72 e Ker{d$ of

6 differ from a boundary, i.e. 35 e C2(M, Z) s.t. y\ = yi + 32^- Because co is

holomorphic, we have JdiS co = 0 ([Sp] 6-5, Thm 6-9) and

I co = I co + I co = I co.

J y\ J Y2 J 828 J Y2

This shows that the integrals of a holomorphic 1-form along two homologous 1-

cycles are equal ([Sp] 6-3, Thm 6-2).

Definition 1.52. Let M be a Riemann surface, co a holomorphic differential 1-form

on M and 6 e Hi(M, Z). For any 1-cycle y representing the class S, the above

allows us to set

ha~La-Note that integration is C-linear in the 1-forms and Z-linear in the 1-cycles (cf [Mir]IV 3).

Taking the homology of a simplicial complex defines a covariant fonctor (betweenthe category of simplicial complices with so-called simpicial applications and the


category of R-modules with i?~homomorphisms). Since we are only interested in

the first homology R-module of Riemann surfaces, we will use the association we

have made above of a piecewise differentiable path to a 1-cycle. Namely, consider

two Riemann surfaces M and N and a holomorphic map 0 : M -» N. Take

then two homologous 1-cycles yx, i = 1, 2, on M and their associated piecewisedifferentiable pathes (ct, I), i = 1, 2. Then the compositions $ o cr for i = 1,2 are

piecewise differentiable pathes on N which correspond to homologous 1-cycles on

N. More precisely, we define

Definition 1.53. Let M, N be two Riemann surfaces and O : M -> N a holomor¬

phic map. Let C = [y] e H\(M, Z) and (c, /) be the piecewise differentiable pathassociated to y. Then $oc defines a class in H\ {N, Z) which depends only on C

and is called the push-forward of C. It is denoted by OtC (or sometimes by $*y

by abuse of notation).

Remark 19. The push-forward of 1-cycles is adjoint to the pull-back of differential

1-forms in the sense that

/ CÛ = / <&*co.

This can be verified by an easy calculation.

Remark 20. If 0 : M —> ./V is an analytic isomorphism, we can define the pull-back of B e H\(N, Z) under O to be the push-forward of D under <ï>-1. Indeed,let's set

<D*D := (O^^D.

One observes that first pushing forward and then pulling back (or the converse) is

the identity.

We end this section by introducing an often used terminology.

Definition 1.54. ([Mir] VIII 1) Let M be a Riemann surface and ß£n[Af] the C-

vector space of holomorphic differential 1-forms on M. Then a linear form À :

Œ^n[M] -> C is called a period on M if there exists 6 e H\ (M, Z) such that,

Vw e ßJjM], l(co) = co.

Je


1.3.14 Sheaves

The notions and notations introduced in this paragraph will hardly be used as theyare here presented. It is however useful to introduce them briefly, in order to be able

to reconize them and translate them into more friendly objects.

Definition 1.55. ([H] III) Let's give us a topological space X and for each open

subset U of X an abelian group 3^(17). If an open subset V is included in U, let's

give us a morphism of abelian groups puv ' 3r(U) -> .F(V) (puv can be viewed

as a restriction and puv(s) f°r s e 3T(U) is therefore sometimes denoted by s\y).

Suppose that

• :F(0) = {0},

• V open subset [/çl, pujj = idp-(jj),

• if W ç V ç U are open subsets of X, then puw = Pvw ° Puv

and V open subset PCX and V open cover {VJ}je/ of U, we have

• if s G 3^(11) satisfies s\v; — 0, Vz e I, then s = 0 and

• for each family {st ; st e F (Vi)} tel satisfying Vi, j G I,Si\v,nv, = Sjl^nv,,there exists an element s G SîU) such that Vf e /, s\yt = S{. (Note that the

previous condition implies the unicity of s.)

Such a data is called a sheaf of abelian groups on X and denoted by T.

Terminology 2. If S7 is a sheaf of abelian groups on a topological space X and U an

open subset of X, then the elements of 3- (U) are called sections of ¥ over U and

.f'(C/) is sometimes also denoted by T(U, 5r). The elements of 3T(X) are usuallycalled global sections of Jr.

Example 2. Let X be an algebraic variety over k. On the underlying topological

space can be defined a sheaf (of rings) by associating to each open subset U of X

the ring of regular functions Ö(U) on U and, for V ç /J open, defining p^y :

Ö(C/) —> O(V) by puv(f) = f\v- Indeed, thefirstthreeconditionsareeasilyverified,whileforthetwolastones,notethatafunctionwhichisdefinedonaunionofopensetsandregular(resp.zero)oneachofthemisactuallyregular(resp.zerotrivialy)onthisunion,becauseregularityisalocalproperty.ThissheafÖiscalledthesheafofregularfunctionsonX.


Example 3. Let X be an algebraic variety over k. To each open subset U of X,

associate the k-vector space QÎU] of regular differential 1-forms on U and, if

V ç V is open, define puy : ^[f/] — ^[V] by puy{co) = co\y which is a

regular differential 1-form on V. The first four properties are easily verified. For

the last one, define s by s(P) := sl (P) if P Vt. This is well-defined, because any

two elements of the family agree on the intersection of their domains of definition.

This makes s to a regular differential 1-form on the whole of U, because regularityof a differential form is a local property. Hence, we get a sheaf on X, which will be

called the sheaf of regular differential forms (or 1-forms) on X and denoted byß (or Qx)-

The key in the two above examples is that regularity (of functions or differential

forms) on an algebraic variety is a local property. Since the holomorphy of functions

and differential forms on a Riemann surface M is also a local property, one can

define with them sheaves on M in a similar way as what we have done here. The

sheaf Q}an (or QlM an) defined by associating to an open subset U of M the C-vector

space Qlan[U] will be called the sheaf of holomorphic differential forms (or 1-

forms) on M.

1.3.15 Complex Tori and Abelian Varieties over C

Definition 1.56. ([Mu] 12.1) Let k be a field (not necessarily algebraically closed).An abelian variety A over k is a projective algebraic variety over k, with a group

law ji : A x A —>- A such that /x and the map v : A —> A given x \-> x~l are

morphisms (of algebraic varieties).

An abelian variety is then a group object in the category of projective algebraicvarieties. The group law actually forces A to be nonsingular.

Definition 1.57. ([BL] Chi) A lattice A in a complex vector space V of dimension

d is a discrete free group of maximal rank.

Equivalently, A is a free abelian group of rank 2d. As such, its generators generateV over R. Moreover, A acts on V by addition, hence the following definition makes

sense.


Definition 1.58. A quotient V/A, where A is a lattice in a C-vector space V, is

called a complex torus.

A complex torus T := V/A can be given the structure of a complex manifold of

dimension dim(V). The complex topology on V/A is obtained by requiering the

quotient map V -> V/A to be continous. T also inherites a group structure given bya holomorphic map T xT —> T. The map T ^> T, x h-» x-1 is also holomorphic.Hence T is a group variety in the category of complex manifolds (such a variety is

called a Lie-group). It can be proved that T is moreover connected, compact and

abelian.

Let A be an abelian variety over C. Then the complex manifold A(C) (see Def¬

inition §1.3.8) is analytically isomorphic to a complex torus V/A, where dim(V)is the dimension of A ([Mu] 12.5). The elements of A are called periods on A.

Conversely, it can be asked whether any complex torus can be realized as an alge¬braic projective variety. The answer to this question is in general negative. In order

to give the necessary and sufficient condition on V/A for it to be embedded in a

projective space, we first recall some notions of linear algebra.

Definition 1.59. ([Ke] 1.2) A Hermitian form H on a complex vector space V is

a pairing V x V —> C such that

1. H is C-linear in the first variable and

2. Vu, w gV, H(v,w) = H(w, v).

This implies

1. VA g C, Vu, w G V, H(y, kw) = XH(v, w) and

2. Vu G V, H(v, i))£l.

Write £" for the imaginary part of H (that is, Vu, w e V, E(v, w) = Im(H(v, tu))),then E : V x V —> Mis a real bilinear form on V with the properties

1. Vu g V, E(v,v) =0and

2. Vu, w g V, E(iv, iw) — E{v, w).


Remark 21. ([Ke] §1.3) H can be recovered from E. More precisely, given a R-

bilinear form E : V x V -> M. satisfying 1. and 2., there exists a unique Hermitian

form H : V x V -» C whose imaginary part is £. (Define H by H(v,w) :=

E(iv, w) + iE(v, w).)

Remark 22. Let H and E be as above and set KerH = {v e V; Vw e V, H(v, w) =

0} and KerE = {v e V; E(v, w) = 0, Vw e V}. Then, we have

KerH = KerE.

Clearly, KerE D KerH. For the converse, take z £ V such that, for each

w e V, E(z, w) = 0 and write H(z, w) = E(jz, w) + iE{z, w). By 2. and the

assumption, H(z, w) = E(—z, iw) = —E(z, iw) = 0. Hence, KerE C KerH.

Consequently, £" is nondegenerate if and only if H is nondegenerate.

Theorem 1.4. ([Mu] 4.1) A complex torus T = V/A can be given the structure ofan algebraic variety if and only if there exists a positive definite Hermitianform H

on V such that its imaginary part Im(H) takes integral values on A x A. In this

case, T can be realized as a projective algebraic variety.

Note that for a complex torus, being a complex projective algebraic variety is equiv¬alent to being an abelian variety (over C) ([Ke] §2.4). Hence, this theorem answers

our question. Given H as in the theorem, the construction of the embedding of T

into a projective space is due to Lefschetz.

Remark 23. By the above two remarks, the data of a positive definite Hermitian

form H on V x V with imaginary part taking integral values on A x A is equivalentto the data of a positive definite alternating E-bilinear form E : V x V ^ M. such

that E(A x A) C Z and E(iv, iw) = iE(v, w), Vu, w G V.

Definition 1.60. ([Mu] 5.9) Such a form E on T := V/A as in the above remark is

called a Riemann form on J or a polarisation of T.

Let's restate the theorem in using this definition:

A complex torus is an abelian variety if and only if it admits a polari¬sation.

We intend now to define an equivalence relation on complex tori ([BL] Chi§2).


Definition 1.61. Let T, T be two complex tori. A homomorphism from T to T

is a holomorphic map T — T which is compatible with the group structures.

Definition 1.62. An isogeny from a complex tori T to a complex tori T is a sur-

jective homomorphism T -> T with finite kernel.

Definition 1.63. Let / : T -> T' be an isogeny between complex tori. We define

the exponent of / to be the smallest integer n satisfying nx = 0, Vx e Kerf.

(Such a n exists and is positive, because Kerf is finite.)

Proposition 1.5. Let f : T -> T be an isogeny of exponent e between complextori. Then there exists an isogeny g : T' -> T, unique up to isomorphism, such that

g o / is the multiplication by e on T and fog the multiplication by e on T.

Since, moreover, the composition of two isogenies is again an isogeny, saying that

the complex tori T is equivalent to the complex tori T if there exists an isogenyfrom T to T' defines an equivalence relation on the set of complex tori. Thus, the

following definition makes sense.

Definition 1.64. Two complex tori T, T are called isogenous, if there exists an

isogeny from T to T. We will say that two abelian varieties over C are isogenous,if their corresponding complex tori are so.

Remark 24. Let A be an abelian variety over k, then any abelian subvariety B of

A has a complement C, i.e. there exists an abelian subvariety C of A such that

A = B + C and B n C is finite. Hence, A is isogenous to B 0 C ([Mu] 12.2). In

the case where A is defined over C, we can view this in terms of complex tori ([Ke]

§9.2). For example, writing X = V/L for the complex torus isomorphic to A(C),then any subspace W of V defines a subtorus Y = W/(W (1 L) of X. Applying the

above to F, there exists a subtorus Z of X such that X — Y + Z and the intersection

Y n Z is finite. Hence, X is isogenous to Y © Z and dimZ = dim Y + dimZ.

The next paragraph provides an example of abelian variety.

1.3.16 Jacobian Varieties over C

We first give the definition as a complex torus of the Jacobian variety of a compactRiemann surface, explain how it can be given the structure of an abelian variety and


finally sketch the relation with the definition of the Jacobian variety of a nonsingular

algebraic projective curve defined over C

Consider a compact Riemann surface M of genus g := g (M). Then from Definition

1.51, §1.3.12, we have dimc(â„[M]) = g and from the comparison of (1.3),

§1.3.9, with (1.9), §1.3.12, we have rankzHi(M, Z) = 2g.

Suppose that g(M) > 0, write a\, ..., aig for a system of représentants of Z-

generators of H\(M, Z) and œ\, ...,cog for a basis of Q\n[M]. Recall (cf §1.3.13)

that the integral / coj does not depend on the choice of the représentant al of the

class [ai]. For i = 1, ...,2g, write

At :=(/ cou..., / cog)eC8Ja, Ja,

and A := ZAX © ... ©ZA2g.

It can be shown (cf [Mu] 3.8) that A is a lattice in V := C8. Hence, the quotient

V/A is a complex torus. The bases can actually be chosen (cf [Fu2] 20d) in order

that, for i = 1, ..., g and j = 1, ..., g, the following hold

cOj = Sij, (1.11)a,

where Stj is the Kronecker's symbol. For the rest of the story, we will suppose that

our systems of generators satisfy (1.11). Then, for i = 1,..., g, Ai is the z'-th vector

et of the standard basis of C^ and Ag+\, ..., A2g determinate A. As A is a lattice in

V, its generators over Z generate V over R. Write Aj, ...,A* for the dual basis of

V* viewed as M-vector space and, for u, v e V, set

s

E(u, v) := ^(A*+g(U)A>) - A*(u)A*+g(v)).i=\

One can show (cf [Mu] 6.10), that this R-bilinear form onVxV satisfies the

conditions for being a Riemann form (see definition in §1.3.15) on the quotientV/A. Consequently, this quotient can be realized as a projective variety and it is

isomorphic to an abelian variety defined over C.

Definition 1.65. Let M be a compact Riemann surface of genus g and A = ZA\ ©

... © ZA2g C C8 be the lattice defined above. Then the abelian variety C^/A is

called the Jacobian variety of M and is usually denoted by Jac(M) (or J(M)).


Now, we would like to give another interpretation of the Jacobian variety of a com¬

pact Riemann surface M in terms of the holomorphic 1-forms on M ([Mu] 6.12).For this, first remark that, if y\, •-, Yig form a system of Z-generators of H\ (M, Z),then the map

#i(M,Z) —> A

yl=[al] h^ Al = (faicü1,...,faicüg)is clearly surjective and preserves the Z-module structures. Since both objects are

free Z-modules of rank 2g, it is actually an isomorphism.

On the other hand, Qlan[M] is a complex vector space of dimension g(M) and so is

its dual (ßlan[M])*. It remains to show how H\(M, Z) embeds in {Qlan[M])* as a

lattice. The map i : Hi(M, Z) -> (ß* [M])* is given by

Yi = [at] i—> [(Yi, } : co h-> l cû\.Ja,

In particular, for i = 1, ..., g, we have i(yt) = co~, because our systems of genera¬

tors satisfy (1.11), and we can apply the so-called Riemann period relations to show

that

2g

3k\, ..., À2g e M s.t. y_, K(Vi, •) = 0 implies Ai = ...= A.2g = 0.

Hence, the vectors (yt, •), i = 1,..., 2g, are R-linearly indépendant and generate

(ß*B[M])* over M. This shows that i(Hi(M, Z)) is a lattice in (^„[M])*.

This implies

Jac(M) ~ (^7î[M])7K#l(M, Z))

and, because fi*„[M] = F (M, &M fln) (cf Terminology 2, §1.3.14), it justifies the

following definition.

Definition 1.66. ([CS] VII by J.-S. Milne §2) Let X be a nonsingular algebraic

projective curve over C. Let X(C) be the associated one-dimensional complex ana¬

lytic manifold (which is compact, cf §1.3.8), film

be the sheaf of holomorphicdifferential 1-forms on X(C) and r(Z(C), fiL(C)ifln) the g(X(Q)-dimensional C-

vector space of global sections of flLC) an.Then the Jacobian variety of X is

defined to be the quotient

r(X(C),^(C)^*M#i(W),z)).


It is denoted by Jac(X) (or J(X)) and is an abelian variety.

Remark 25. Remembering that ßl(C) an[X(C)] = &[X] (cf Remark 18, §1.3.12),we can also write the Jacobian variety of X as

Q^X]*/i(Hi(X(C),Z)) .

Remark 26. According to Definition 1.54, §1.3.13, the periods on a Riemann sur¬

face M are the linear forms in ß*n[M]* of the form (y, } for one y e H\{M, Z).Such elements are also the periods on the abelian variety Jac(M) (cf §1.3.15).

Remark 27. Let M be a compact Riemann surface of genus g > 0 and Po a fixed

point on M. Let further œ be a holomorphic differential 1-form on M and P any

point on M, then the integral fp co depends on the choice of the path on M goingfrom Pq to P. But since the difference of the integrals along two such paths is the

integral along an element of H\ (M, Z), we have a well-defined map

 C*/A = Jac(M)p p

P i—> (f m,..., f (Og),Po Po

where A is the periods lattice and co\, ..., cog form a basis of Q,\n[M]. One can show

that (p is an embedding and that the induced map

<p* : nlan[Jac(M)] -> tl]inW]

is an isomorphism.

In the case where M has genus 1, it can be shown that <p itself is an isomorphism. It

follows that each compact Riemann surface of genus 1 is isomorphic to a complextorus C/A and thus to an elliptic curve over C (compare with §2.2.1).

1.3.17 Algebraic Appendix

This paragraph concerns exclusively linear representations of finite groups. We justintend to recall some definitions and properties and give an example that will be

used in Section 1.4.


Definition 1.67. ([Se2] 1.1) Let G be a finite group and k a (commutative) field.

Then a linear representation of G is a finitely dimensional k-vector space V to¬

gether with a group-homomorphism

p : G — GL(V)

s i—> ps.

This is actually equivalent to the data of a linear action

GxV — V

(s,v) i—> ps(v).

In particular, V is a G-module. The dimension of V over k is also called the di¬

mension (or degree) of the representation.

Definition 1.68. ([Se2] 1.3) A linear subrepresentation U of V is a linear sub-

space of V, which is invariant under the action of G. Equivalentely, U is a sub-&G-

module of V.

Definition 1.69. A linear representation V of G is said to be irreducible, if V ^{0} and the sole submodule of V are V and {0}.

From now on, we will suppose k = C, because the following is not true for any field.

Namely, k may have to be algebraically closed or of characteristic not dividing the

order of G.

It can be shown ([Se2] 1.4) that the complement as a linear subspace of any sub-

CG-module of V is G-invariant. Hence, any linear representation admits a decom¬

position as a direct sum of irreducible subrepresentations.

Let CG be the C-vector space on the set {es}seG- Then, setting s et = est for

s,t e G and extending C-linearly to CG, makes CG into a linear representation of

G. It is called the (left-) regular representation of G.

One can show ([Se2] 2.4) that each irreducible CG-module U is isomorphic to an

irreducible factor in the decomposition of CG. More precisely, if d is the degreeof U, then there are d2 factors isomorphic to U. This shows, in particular, that the

isomorphy classes of irreducible CG-modules are in finite number and that

\G\ = J2 dim(U)2. (1.12)Uirred

CG—module


Now, given any linear representation V of G, its decomposition in irreducible sub-

representations is not necessarily unique. But ([Se2] 2.6), if you take one such

decomposition, say V — U\®...®Utl, and write W\, ...,Wh for the irreducible CG-

modules and, moreover, for each i e {1,..., h} such that there exists k e {1,..., h}with Uk ~ W(, you write Vi for the direct sum of the Uk's that are isomorphic to

W;,then

v = ©v,.i

This decomposition does not depend on the choice of the initial decomposition and

is unique.

Definition 1.70. ([Se2] 2.6) A linear representation which is the direct sum of

pairewise isomorphic subrepresentations (such as Vt) is called isotypical and the

decomposition of V in isotypical components is called canonical.

To a linear representation V over C of a finite group G, one can associate a function

G —> C which characterizes the representation.

Definition 1.71. ([Se2] 2.1) The character x °f a linear representation p : G —>

GL(V), s h» ps, is the function

G —» C

s i—> Tr{ps).

A character of a linear representation of G is also called character of G and the

dimension of V the degree of x •

One can show ([Se2] 2.3) that two linear representations of G are isomorphic if and

only if their characters are equal.

Definition 1.72. A character of G is said to be irreducible, if it is the character of

an irreducible representation of G.

Remark 28. Because the trace is invariant under conjugation, so are the characters.

Such a function is called central. It can be shown ([Se2] 2.5) that the irreducible

characters of G form a basis of the C-vector space of central functions on G. This

shows that the number of irreducible characters of G equals the number of conju-

gacy classes in G.


Let h be the number of conjugacy classes in G and xi, ••> Xh the irreducible char¬

acters of G. Write dt for the degree of Xi •Then (1.12) can be written as

h

\G\ = J2dl (1.13)

Suppose now that G is abelian. This is equivalent to the fact that each conjugacyclass of G contains exactely one element. In this case, their number equals \G\.

Together with (1.13), we get

Proposition 1.6. ([Se2] 3.1)

G is abelian <=>- Each irreducible linear representation of G has degree 1.

Corollary 1.7. The irreducible characters ofan abelian group are homomorphisms.

Example 4. Consider the group ji^ of complex iV-th roots of unity. Let jx^ be

generated by the primitive iV-th root £>/> X be an irreducible character of p,^ and

set x(£iv) = v. Since x is a group-homomorphism, Vk e Z, x(£a>) = y/: and,

in particular, vN = 1. Hence, v is also a JV-th root of unity and the N irreducible

characters xo, , XN-l of /^n are given by

X/îVSyv) = SA? !

where n, k e {0, ...,iV - 1}.

1.4. Abelian Varietiesfor Appell-Lauricella Hypergeometric Series 53

1.4 Abelian Varieties for Appell-Lauricella Hyperge¬ometric Series

1.4.1 Motivation and Structure of the Section

The Appell-Lauricella hypergeometric series F\(a, hi, ..., br; c; A.2, -, K) in the

r — 1 variables À2, ..., Ar (cf Section 1.1) has the integral representation

1— / x-c+^'bJ(x - i)c-a_i r\(x-*-irb,dx.i,a)Ji

nB(c-a,

.., U1

J =2

The above integrals converge for Re(c) > Re (a) > 0, but it is sufficient that

rr

a, —c +

in order to replace the above quotient up to an algebraic factor by a quotient of

periods on algebraic curves. The algebraic multiple of the integral on the numerator

is an integral

J x~ß0(x - 1)-^ pj(jc - Xj)~ß'dx, (1.14)

where y is the lifting on the following algebraic curve of a Pochhammer cyclearound 1 and 00

r

yN =xNß0(x_1)Nßl Y[(X-Xj)NlJ->. (1.15)

7=2

Here, N is the smallest common denominator of

Po = C-Ylj=2bj,jjl\ = 1 + a — c,

fij = bj, for j = 2, ...,r.

The integral (1.14) can be written as J y and is the integral of a differential 1 -form

on the projective algebraic curve defined affinely by (1.15). If, for k = 0, ..., r, we

write Ak := Nßk, then the condition that ßo, , ß>- £ % implies that

N\A0,...,Ar.


Moreover, since Ylk=o Ak = N(a + I), the condition a £ Z implies

r

k=0

In the next paragraph, we first show that we can in fact suppose the exponents

Aq, ..., Ar to be positive. Then in §1.4.3 we define the family of curves for which

we will construct abelian varieties of dimension depending only on N. Since we

will start from a singular curve Cjv, we have to desingularize it. This is done in

§1.4.5 after we have determinated the possible singular points of Cm (cf §1.4.4).In §1.4.6, we calculate the genus of the nonsingular curve X^- In order to find a

basis of the vector space Ql [X^] of regular differential 1-forms on X^, we use the

fact that the group //,# of JV-th roots of unity acts on X^ and therefore on &[Xn].These actions are defined in §1.4.7. In §1.4.8, we calculate a basis for £21[Xn] and

the dimension of the isotypical components Vn for the action of ß^ on ^[X/v], first

in the general case n g {0,..., N— 1}, then in the special case where (n, N) = 1. For

this last case, we show that the sum dim Vn + dim V/v_„ is constant. In §1.4.9, we

explain why we will only be interested in the V„'s with (n, N) = 1 and in §1.4.10

define with them an abelian subvariety of the Jacobian variety of X^, which will

be called the "New Jacobian" of X^. Its dimension is calculated as a function of N

and r.

1.4.2 Reduction

The curve we will be interested in is the desingularization X'N of the projectivecurve C'N with affine equation

yfN = t\(x>-K)A',z=0

where N e N, Aq, ..., Ar e Z. In this paragraph, we want to show that, to our aim,it is sufficient to suppose that Aq, ..., Ar are positive. The case A( — 0 is implicitelyexcluded, because we do not want to talk about a factor that does not appear.

Indeed, for each i e {0, ..., r}, write A{ = k(N + 77, where 0 < rt < N — 1 and


ki e Z. Note that if At < 0 then ki < 0. Hence, the equation reads

/* n& - Xi)~kiN=n & - ^ n & -Xi)AiA,<0 A;<0 A,>0

If we denote by C^ the projective curve with affine equation

yN= n^-^ n^-Â,<0 A,->0

we have a map p : C'N -> C/v given by

(*', /) h> (*', / Y\Ai<0{x> - Xtrk') =: (*, v)

OO h-» oo.

It is well-defined, because if (V, y') e Cjy, then

?*=(y n <*' - ^r*o*=n(x/ - A-')r' n<*' -ÂiA,<0 A,<0 A,>0

and (x, y) e CV

A, < 0 implies —£j > 0 and p is a morphism. Tt has a rational inverse given by

(x,y) h> (x,yY\A.<0(x - ki)ki). Write tt' : X'N -> C'N and re : XN -+ CN

for the desingularization maps and remember that they are birational morphisms.Then p o it' is a birational morphism from X'N to Cjy and there exists a unique

isomorphism p : X^ —> X^ such that the following diagramm commutes

X'N > Xn

C'N > Qy.p

Consequently

ß*(&[xN]) = n1[x,N]

and the existence of an abelian subvariety of Jac(X^) on which an integral / co,

co ^[Xat], lives as a period implies the existence of an abelian subvariety of

Jac{X'N) of the same dimension on which / p*co lives as a period.

From now on we will then suppose Aq, ..., Ar > 0.


1.4.3 Definition of the Family of Curves

In this paragraph, we give the precise definition of the family of curves that will be

considered in this section.

Let r e M U {0}, N,A0,..., Ar G N and

r

N\AQ,...,Ar,J2Ak-k=0

Note that, in particular, N ^ 1, Y^'k=o Ak- Let give us also Xq, ...,Xr e C such

that, Vz, j e {0, ..., r] with i ^ j, A; ^ Xj. We define Cm to be the projective planecurve with affine equation

r

yN = Y\(x-Xi)A', (1.16)

and X?j to be its desingularization (cf §1.3.7). C^ and X^ are projective algebraic

plane curves defined over C.

In order that the set of complex points of Xm form a Riemann surface, we have to

suppose Zyy to be irreducible, that is

(N,A0, ...,A,-) = 1.

This hypothesis will also be used in § 1.4.8.

To get the projective equation of C;v, we set x := ^-, y := ^ and substitute this in

(1.16). We get:

. Case 1:N- £Lo Ak > 0; CN : xf = jc"_EU A*U'î=oî ~ ô)A<

• Case 2:N- £Lo Ak < 0; CN : x?x~N+n=°Ak = JlUî " XÂ'

By definition, the points at infinity are those with coordinate xq = 0. In each case,

there is only one point at infinity. It is (0 : 1 : 0) in the first case and (0 : 0 : 1) in

the second one.

Remark 29. The case N — Yl'k=o Ak = 0 is excluded by our hypotheses.


Recall that each time we need to know the local behaviour of Cm at an affine point,it is sufficient to consider the affine equation of Cm and the affine coordinates of

this point.

Remark 30. When we do not wish to distinguish the cases N — X^=o Ak > 0 and

N — Yl'k=o Ak < 0, the point at infinity will be denoted by oo.

1.4.4 Singular Points on Cn

Recall that the singular points are those whose coordinates anihilate all the partialderivatives of the polynomial defining the curve (cf § 1.3.7).

• Affine Singularities: f(x, y) :— yN — ]~]]=0(x — kt)A'

§1,u*>)>)

= -Zj^rAj (x -kj)A'~in;=o(^ -*.-r« = 0

O 3j e {0, ..., r} s.t. x = kj and A} > 1,

Y;(x, y) = NyN~l = 0 4> y = 0 (because N > 1 per hypothesis).

Since for (x, v) e Cm, x = kj implies y = 0, we have

(kj, 0) is singular <& Aj > 1

Singularity at Infinity: Calculating as above, but on the projective equations,one finds in both cases

1.4.5 Structure of the Desingularization

In this paragraph, we construct the desingularization Xm of Cm by gluing local

desingularizations. These are constructed explicitely in the first subparagraph. By


the way, we get the number of preimages on X^ of each potentially singular point of

Cn under the desingularization map tc.This will be used in the computation of the

genus of Xn in § 1.4.6. In the second subparagraph, we explain how the nonsingular

projective curve X^ can be constructed by gluing these local desingularizations. In

the last one, we calculate the compositions of jr with local parametrizations of Xjy

at the 7i-preimages of the potentially singular points of CV- These will be used in

§1.4.8 to calculate the order of the pull-back on Xm of some differential forms on

Cn-

Local Desingularizations

Let P be a potentially singular point of C^ (cf §1.4.4). We will work locally at P

by restricting to an open neighbourhood of P isomorphic to an affine curve.

• Above Pj := (1 : kj : 0)

In the neighbourhood of an affine point, we have the classical isomorphism

C]V - {OO} -> Caff

(*o:*i:*2) » (^,g).

It induces a morphism ko ' Caff —>- Cn given by (x,y) \-> (1 : x : y).

Setting x := ^-, y := —, we recover the affine equation of Cjv (cf § 1.4.3):

Caff- yN = Y\(x-Â'-i=0

Let's fix j g {0,..., r] and work locally in a neighbourhood of (XJt 0) on

which 8j(x) := ni#J(* - K)At Ï 0. Set

N,

A,

N' := — and A' :=J

(N,Aj) (N,A})

Then there exist n,m e Z such that

nN' + mA' = 1


and we have

,JV'_ fv i .\A'

(x, y) G Caff

Remark that

y"=

(x-

Xj)A u and

u(N.Aj) =gj(xy

ymN\x-\j)nN' = {x-\j)um

and that u = yN (x — Xj)~A'. Hence, if we set z '= ym{x — Xjf and define

Xj := {(x, u, z) e C3; zN' = (x - Xj)um, u{N'A^ = gj(x), gj{x) # 0}

and Caffj := Caff - {(x, y) e Caff, gj(x) = 0}. Then the rational map

vi : c«ff-i —y XJ

(x,)0 i-^ (x,yN\x-kj)-A',ym(x-kj)n)

becomes a morphism on the open dense subset Caff '.= Caffj — {(Ay, 0)} of

Caffj- This morphism has an inverse given by the morphism

{x,u,z) i—> {x,unzA).

In particular, Xj is a birational morphism, which restricts to an isomorphism

from Xj := Xj - (tT1«^, 0)}) to Caff = Caff - {{Xk, 0); k = 0, ..., r}.

Moreover, since Caff is isomorphic to C/v —{oo} under kq, Caff is isomorphic

toCiv - {Po, ..., Pr.oo} and

i7- -^ Cjv-tPo,...,^,«)}.

In particular, Xj is birationally equivalent to C^ under the morphism itj :=

/Co O Tj.

Remark31. The point (Ày, 0) e Caffj has exactely (JV, Aj) preimages un¬

der Tj, which are the points {Xj, u, 0), where « runs among the (jV, A;)-throots of gj{Xj). (They are distinct, because gj{Xj) ^ 0.) Pj e C^ has then

also {N, Aj) itj -preimages on Xj.


Remark 32. Xj is nonsingular. Indeed, calculating the Jacobian matrix of

Xj, we get

um g'.(x)

mix - Xj)um~l -(N, Aj)u^N'A^-1~N'zN'-1 0

Remember that u j^ 0 on Xj. If x = Xj, then the above square looks like

( % *0 ) and has rank 2. If x / Xj, then zÔ and the low square looks like

^0 qI and the matrix has rank 2.

Above Infinity

- Case h N -J2k=oAk > 0

In this case, the projective equation of Qv is

x» = xH-ZA>f\{xl-kixo)A-/=0

and the point at infinity has coordinates (0 : 1 : 0). We choose the

neighbourhood of (0 : 1 : 0) on Ov, on which x\ ^ 0, and have the

isomorphism

cN n {xi ^ 0} -^ Coo!

{xq-.xx-.X2) ^ (BL,ZL)=:(x,y).

Its inverse is given by K\ : (x, y) h-> (x : 1 : y). Remark that the

affine potentially singular points with coordinate x\ ^ 0 also lie on

Cm n {xi ^ 0}. In the coordinates (x, y), the equation of Cî is

Cooi: yN=xN-ZA*fl(l-X.ix)A',i=0

the point at infinity is (x, v) = (0, 0) and the affine potentially singularpoints are the points (^, 0), for each k e {0, ...r} such that Xk ^ 0. If

we set h(x) := U'i=o(l - hx)Al,

N,

N-TAk

N' := — and A' :=^ k

(N,N-J2Ak)'

(N,N-J2Aky


there exist n, m g Z such that

nN' + mA' = 1

and we have

(x,y) G Cooi

Note that

yN = xA u and

M(N.W-£A,)=/z(x).

and that u = yN'x~A'. Set further z := ymx'\

XoDl :={(x,u,z)g£3; zN> = xum, u^N~Âk) =h{x),

h(x) Ï 0}

and C^j := Cooi — {(x, y); h(x) — 0}. Then we have a rational map

vool : ôol * Xool

(x,y) h^ (je,yv'jc-A',;ymJcn),

which restricts to a morphism on the open dense subset Cool '= Côi ~~

{(0, 0)} of C^j. This morphism has an inverse given by the morphism

ôol • ôol ^ ôol

(x,u,z) I—> (x,zAun).

Tool is then a birational morphism and restricts to an isomorphism of

Xooi := Xçqi - t^jKO, 0)} to Cod. Remembering that Cool is iso¬

morphic to Cn H [x\ ^ 0} under k\, then Cooi is isomorphic to Cm —

{Po,..., Pr, oo} and so is Xooi, he.

Xool — Cm - {Po, ••> Pr, oo}.

In particular, X^d is birationally equivalent to Cm under the morphism

ôol := K\ o Tool-

62 1, Abelian Varieties Associated to Hypergeometric Series

Remark 33. The point at infinity, which has coordinates (x, y) = (0, 0)on Cooi, has (N, N — J2 A-k) preimages under Tœ\. They are (0, u, 0),where u is a (N, N — J2 Ajt)-th root of h(0) and h(0) =£ 0. This impliesthat the point at infinity on Cyv has also (N, N — J2 Ak) preimages on

Remark 34. One verifies that Xcoi is nonsingular.

Case 2:N- Y?k=o Ak < 0

In this case, the projective equation of Cm is

x?x-N+Âk = Yl(Xl-xlXoy

and the point at infinity (0:0:1). We choose the neighbourhood of

(0 : 0 : 1) on C/v, on which X2 # 0. On this neighbourhood, there is

no other potentially singular point than (0 : 0 : 1), because they all have

coordinate xi = 0. We have the isomorphism

cNn{x2^0} — Cco2

(X0-.Xi-.X2) H-> (J,J)=:(ij),

with inverse/C2 : (x, y) \-> (x : y : 1). In these coordinates, the equationof Coo2 is

C002: x-N+ZAL = i\(y-ktx)A<i=0

and the point at infinity (x, y) = (0, 0). But the equation in this shapeis not easy to handle. We will see that after having blowed up the point(0, 0) on Coo2, everything becomes easier. In order to do this, we will

use the expressions of the blow-up map in local coordinates as given in

§1.3.6.

In the first coordinates' set, the point (0, 0) has no preimage on the

preimage of Cœ2 bereft of the exceptional divisor. In the second co¬

ordinates' set, the preimage of Cqo2 under the map (uv, v)

1.4. Abelian Varieties for Appell-Lauricella Hypergeometric Series 63

is given by

riv+£a^-jv+ea* = „EAt j-[;.=o(i _ à.m)a,

v = 0 (exceptional divisor), or

cTO2:^+EA* = ^n;=o(i-^)A<

and the preimage of (x, y) = (0, 0) is (u,v) = (0, 0).

We can apply to C'2the same proceedure as in the other cases, though

it will be slightly more technical. As usual, we begin by setting h(u) :=

n;=o(i-ô\

N':=N

(N,-N + -£Ak)and A' :-.

(N,-N + J2Ak)

and letting n, m e Z be such that nN' + mA' — 1. Then («, v) e Cîmplies

A' N'

w(N,-N+Y:AÛ=hiu)

h{u) ^ 0.

Note that unA'vmA' = vwn and that w = uA'v"N'. Set z := unvm and

define

Xoo2 := {(«, v, w, z) G C4; zA' = vwn, ww~Ât) - ä(k),

h(u) ^ 0}.

Then we have a rational map

ôo2C'oo2

-* zoo2

(«, i>) i—> (u,v,uAv ,unvm),

which restricts to a morphism on the open dense subset C^ — {(0, 0)}

of C^2. This morphism has an inverse given by the morphism

Lco2 Xco2 + C~2

(w, u, tu, z) i—> (wmzN , v).


Hence, r'2

is a birational morphism and restricts to an isomorphismof X0O2 := Z002 - {(r^)"1«), 0)} to C^2 - {(0, 0)}. Rememberingthat the blow-up map cp : C

2—> Cî is a birational morphism and re¬

stricts to an isomorphism on C^ —{^_1{(0, 0)}} and that <p~l{(0, 0)} =

{(0, 0)}, we get a birational morphism

ôo2 '= (f o roo2 : Xoo2 —> Coo2,

which induces an isomorphism from Xœ2 to Coo2 := Cool ~ {(0, 0)}.

Now, since C^ is isomorphic to C/v — {Pq, ..., P,-} under k%, C^ is

isomorphic to Cjv — {ô> ••-, Pr, oo} and so is Xoo2, i.e.

ico2-^ CN-{PQ, ..., Pr, OO}.

The birational morphism from X^ to C# is given by 71002 := &2 ° ^002-

Remark 35. Xœ2 is nonsingular and the T^-preimages of (0, 0) e Cœ2

are (0, 0, w, 0), where w runs trough the (N, —N + ]T Ajt)-th roots of

h(0). Since /z(0) ^ 0, their number is (N, -N + J2 M) = (N,N -

LA*).

Construction of Z^y by Gluing

We refer here to the construction described in [Sh2] V.3.2. Let Xqq resp. 71,30 denote

Zool resp. JTooi in the case N — J] A^ > 0 and X^ resp. 7^2 in the case N —

EAit<0.

Remember that, for each j e {0, ..., r, 00}, the morphism 7T;- : X^ -> Cat restricts

to an isomorphism of the open dense subset Xj of Xj to C^ — {Pq, ..., Pr, 00} and

that Xj and Cn are birationally equivalent. Then one can define an equivalencerelation on the disjoint union U/6{o r0o\Xj by setting, for Qj G Xj, Qk g Xkwith j, k e {0,..., r, 00) and j ^ k,

Qj~Qk^nj(Qj)=jrk(Qk).

Moreover, the functions tïj, j e {0, ..., r, 00}, induce a well-defined function 71 on

the quotient X := ]Jje{0 roo}^/7~ by setting, for C X and gy- G Xj with

[£j] = e,

7T(e):=7r,-(ß/).


By defintion of the equivalence relation, this is indépendant of the choice of the

représentant of the class G.

On the set X, we have the quotient topology and can define a sheaf induced from the

sheaf of regular functions on each Xj, for which n is again a birational and finite

morphism. This implies that X is again a projective curve. Since X is moreover

nonsingular, because so is each Xj, and, as we have seen, birationaly equivalent to

Cn, it provides a model of the desingularization of Qv, hence is isomorphic to X^.The desingularization map is given by the map n such that jt\xl = Jtj for each j.Another isomorphic construction is given in [Shi] 11.5.3 Thm 6, Thm 7.

Since, to our purpose, we only need to know the desingularization up to isomor¬

phism, X will be identified with Xjy in the following.

Compositions of jt with Local Parametrizations of Xjy.

Because the restriction of it to X^ — {7r_1({Po, , Pr, oo})} is an isomorphismto Cm ~ {Pq, ..., Pr, oo}, we only need to know the compositions of n with local

parametrizations at the points of tv~1{{Pq,..., Pr, oo}), which are isolated on X^.

• AbovePj = (i : Xj : 0)

Fix j {0, ..., r) and remember that the local desingularization above Pj is

Xj = {(x, u, z) g C3; zN' = (x - kj)um, M(iV'A^ - gj{x), gj{x) Ï 0}.

We have then the composition

Ttj = kq o Xj : Xj —> Cjv

(x, u, z) i—> (1 : x : unz ')•

Recall that Xj is nonsingular and open in an affine variety, hence each pointhas a neighbourhood for the complex topology which is isomorphic to an

open neighbourhood on C (cf §1.3.8). Choose a complex open neighbour¬hood Uj of s = 0 in C on which gj(sN + kj) / 0. Then the image of la¬under s \-> gj(sN + Xj) is included in C bereft of a half- line through 0.

Thus, branches of roots of gj(sN + Xj) can be well-defined as holomorphic


functions of s on Uj. Then choosing fixed branches, we have a well-defined

holomorphic function

(Pj :sh^ (SN' + XJ,gj(sN' +XJ)w^,sgJ(sN' +\j)%)

from Uj to X}. Indeed, u{N-A^ = gj(sN' + kj) = g}(x) and

m

= (^ +X.J-),J)gJ(SN + kJ)(NA>) = (x-Xj)um.

On the image of <pj, we have a well-defined holomorphic inverse map

m

(x,u,z) i—> zgj(x) jv.

Hence, cpj is an analytic isomorphism and a local parametrization of Xj at

i

cp(0) = (kj, gj (kj)(N A,) ,0), one of the 7r-preimages of Pj <s C^. Remark

again that the choices of branches for the (N, A7)-th root of gj (sN +kj) are

in bijective correspondance with the 7T-preimages on X^ of P}.

The expression of tc in this local parameter s at each tc- preimage of Pj is

given by the composition tCj o cpj ; JJ} —> C/v

s \--+ (ll^V+A; -.s^^gjis^^+kj)^) (1.17)

Above Infinity

- Case h N -J2'k=0Ak > °

We are looking for an analytic parametrization of Xî at the points(0, u, 0), where u^N-N~Âl^ = h(0). Let's choose a complex neigh¬bourhood Uoo of s = 0 in C on which h(sN ) j^ 0, N' being here

(TV N-Y A )•^n sucn a neighbourhood, we can define roots of h(sN')

as analytic functions of s. Fix an 7V-th root h(sN )/v. Then, for each

branch of the (N, N — ^ A^)-th root of h(sN'), the map

«Pool : Uqq —> Zooi

1.4. Abelian Varietiesfor Appell-Laurieella Hypergeometric Series 67

is a well-defined analytic map such that </?ooi(0) = (0, u, 0), where u is

the corresponding (N, N — ^ A^)-th root of h(Q). <poo\ has an analyticinverse on its image, which is given by

_

m

(x, u, z) i—> zh(x) N.

This shows that cpî is a local parametrization of Zooi at the preimagei

(0, h(0)<-N-N~Âk\ 0) of co. Since ttî = kx o xœi : Xœi -» CN is

given by (x, w, z) h> (x : 1 : zAV), tiî o ^j : Uoo —> Cn is given

by

S h

/v

1W-E-lft /V !

—^ (jCWW-EAfc) 5 (A' A'-E A^ /j (^ (JV W-E ^) ) n ) (1.18)

and is a local expression of tv at each point on X^ lying above (0 : 1 : 0).

- Case 2: N - J2l=o Ak < °

On C/oo defined as in Case 1, we have a well-defined holomorphic map

^002 : ^co —> X,oo2-H+Y.Ak

1— (sN>, s^' L"+Z*k)h(sN'ryr, h{sN') <-N--N+Âk).

,,,nN'-l

sh(sN) NA> ),

for a fixed choice of the branch of the roots ofh(sN ). On its image, this

map has an analytic inverse given by

l-niV'

(u, v, w, z) 1—> zh(x) NA>.

Each choice of the branch of the (N, —N + ]P A^)-th root of h(sN )

corresponds to a it -preimage of (0 : 0 : 1) at which <poo2 is a local

parametrization. The composition of (p^ with tï equals k% o <p o x'2

o

<Poo2 : Uoo -^ Cn and is given by

s \-

HAk ~N+j:Ak N,

— (j(W--W+EAt) _s(N.-N+J2Ak) :h(s(N--N+Â^)N) (1.19)

1.4.6 Genus of XN

The aim of this paragraph is to calculate the genus g[X^] of Xm- The curve Xîs irreducible, nonsingular and defined over C, hence the set X^ (C) of complex


points of Zyv can be given the structure of a Riemann surface, which is compact (cf

§1.3.8).

Since g(X^(C)) = g[X^] (cf (1.10) in §1.3.12), we can use the Hurwitz formula

(Proposition 1.3, §1.3.9) to calculate x(XN(Q) and then apply (1.9), §1.3.12.

Consider the projection p : Cn —> P^ given by (xo : x\ : x%) i—> (xq : x\) and

compose it with the desingularization map tx : X^(C) —> Cm- The composition

v := port :XN(Q — Flc

is nonconstant and regular (hence holomorphic) and we can apply Hurwitz formula

to it (Xjv(C) and P^ being compact Riemann surfaces). The degree of v is TV,because each affine point (x, y) G CV with x ^ Xq, ..., Xr has N distinct preimageson Xyv corresponding to the iV-th roots of n)=o(x ~ î)A'

It remains to calculate the ramification indices.

Each point Q g X^(C) such that tc(Q) =: P is nonsingular is a regular pointof the covering v. Indeed, if P e C^ is nonsingular and P = (x,y) (resp.co), then y (resp. x\ and xn) can be written as a function of x (resp xq) in

a neighbourhood of P. tx being locally an isomorphism at Q, x (resp. xq)can also be taken as a local parameter of Xyv(C) at Q. In this parameter, tx

is given byx h> (x, y(x)) (resp. by xo h> (xo : xi(xo) : X2(xo))) and v byx h> x (resp. xo h>- xo). Hence rv(Q) = 1.

For j e {0, ..., r), let Qj be one 7r-preimage of (Xj, 0) e C/y. Rememberingthe composition (1.17) of tx with a local parametrization of Xpj at Qj givenin §1.4.5 and composing it with p, we get

s^ (1 :j(Wj) +à;).

If we now choose the chart (xo : xi) i->1 '

on {(xo : x\) G Pi,; xo ^0} and compose it with the above map, we get the expression of v in local

coordinates as

s i—> s(N.Aj) Thisshowsthateach7r-preimageof(Xj,0)hasramificationindexequalstoiV(AU./)"


• In the cases of the points above oo, we will choose the chart (U\, i/0 on P^,where U\ := {{xq : xi) e P^; x\ # 0} and \jr : {xq : x\) i-> f~-

- Casel: N -J2'k=oAk > °

The composition of p with the composition (1.18) of it with a local

parametrization of X?j at each preimage of oo reads

Composing it with t//-, we get the expression of v in local coordinates at

each 7T-preimage of oo as

- Case 2: N - Y!k=ù Ak < °

In this case, we have to be a little more careful, because the map p is

not defined at oo. If only to consider momentarily the restriction of p

to the punctured Riemann surface Xjv(C) — {7r_1{(0 : 0 : 1)}}, we can

suppose s ^ 0 and consider the composition of (1.19) with p, which is

Y,Al -N+Y. Aj N

If s tends to 0, y o ç?oo2(^) tends to (0 : 1), the point at infinity in P^.Hence, we can extend the map continously by setting 0 t-> (0 : 1). The

composition with \J/ of the extended map is

s |—> SW^N+ZÄÖ

and the ramification index of each point lying above oo is N/(N, —N

+ £A*).

Since (N, —N + J2Ak) = (N, N — £ At), we have found in both cases

that each it -preimage of oo has ramification index equals to (N, N — £ Ak)under v.

Summary 1.

point P of Cjv nb of 71-preimages Q rv(Q)

(^,0), je{0,...,r} (N, Aj)N

(N,A,)

oo (N,N-J2'k=0Ak)N

W,N-E'**AÛ

other points 1 1


Theorem 1.8. Let X^ be the desingularization of the irreducible projective alge¬braic plane curve Cn defined over C by the affine equation

.N Y$x - kiY

i=0

where Xq, ..., Xr e C are such that, Vi, j e {0, ..., r} with i ^ j, X( =^ X} and

N, Aq, ..., Ar eft such that

N ^ Âkand(N,A0, ..., Ar) = 1.

k=0

Then the Euler characteristic ofX^(C) is given by

X(XN(Q) = -rN + (N, N-JÂk) + JîN, Aj)k=0 j=0

and the genus ofX^ by

Ir r

g[XN] = g(XN(£)) = 1 + -(rN -(N,N-J2 Ak) - £](#, A;))k=0 j=0

Proof We apply the Hurwitz formula (Proposition 1.3, §1.3.9) to the covering map

y = p on : XN(C) —> P^,

where it : Xn -> Cm is the desingularization map and p : Cm -> P^ the projectiongiven by (xq : x\ : x%) \-> (xq : x$. As seen above, it has degree N and the only

possible ramification points lie above oo e P^ and (1 : Àj), j e {0, ..., r}. Usingthe ramification indices calculated above, the number of preimages calculated in

§ 1.4.5 (all recalled in Summary 1) and the fact that x (P^) = 2, we get, by Hurwitz's

formula

X(XN(Q) = 2N- (N, N-J2Ak)((NNÂk) -1) -^(tf, AjX-^-l)= -rN + (N, N-J2Ak) + Erj=o(N, Aj).


To get the second formula, remember that g[XN] = g(XN(Q) (by (1.10), §1.3.12)and that x(Xtf(C)) = 2 - 2g(XN(Q) (cf (1.9), §1.3.12). Hence

S(Xjv(C)) = l-\x(XN(C))= 1 + i(rjv - (TV, TV - ELo^) - E7;=o(^- A;')).

D

1.4.7 Actions of fi^

Let /x^ be the group of complex N-th roots of unity. In this paragraph, we define an

action of ßM on Xm and show how it induces a linear action on the C-vector space

^POv] of regular differential 1-forms on Xpj.

For t, e ßN and an affine point (x, v) e Cm, define

^(x,y):=(x,r1y)-

Further, set f • oo = oo, V£ e /x,y. Because (£ _1)N = 1, ^(ij)eCiv and this

is an action, because ßM is abelian and 1 acts as the identity. Moreover, since ßM is

included in the definition field C of Cm, for each Ç e ßM, the map

<Pt, ' Cn —> Cm

(x,v) i—> Ç-(x,y)

OO I—> oo

is a morphism of algebraic varieties.

Now, we want to extend this action to an action on Xm- Remember that the desin-

gularization map tc : Xm —> Cm restricts to an isomorphism on the dense subset

Tt~l{C}ft8) of Xm, where C'^8 is the set of regular points in Cm- Let P e Xm, if

Tt(P) is regular, set

f P:=Jt~l^-Tt{P)).

If P e XN is such that tc(P) - oo or 3; e {0,..., r} with 7r(P) = (Ay, 0), set

Ç P := P.


Note that, if (Xj, 0) (respectively oo) is regular on Cn, this last definition is coherent

with the above one. This defines an action on X^.

For £ e fix, set

Or : Xn —> Xjv

P l—> Ç P.

Then <&f makes the following diagram commute

X# > Xn

(1.20)

C^v > Cm-n

Because cpç and tv are morphisms, $^ will also be a morphism.

Let co be a regular differential form on X^. Since Of : X^y -> Xjy is a morphism,the pull-back <ï>îco is again regular on Xa? (see the end of §1.3.11). Hence, the

following map is well-defined

fiN x ^[XN] —> &[Xn]

(£, co) i—> *co.

It defines an action of fi^ on ^[X^], which is linear, because, for every £ e /x^,

the map ^[X^] —> Ql[X^], to i-> O*^ is linear. Further, since the C-vector

space ^[Xjv] is finitely dimensional (cf §1.3.11), it furnishes, together with the

above action, a linear representation of /x^ (in the sense of § 1.3.17).

As seen in § 1.3.17, such a linear representation admits a decomposition in isotypical

components (the so-called ''canonical" decomposition). The isotypical component,which is the direct sum of irreducible representations of character Xn for one n e

{1, ..., N] (cf Example 4 in §1.3.17), will be denoted by Vn. In these terms, we can

write the canonical decomposition of ^[X^] as

n1[xN] = Q)vn, (i.2i)

where the sum is taken over the indices n in {0, ...,N — 1} for which dim Vn > 0.

In the next paragraph, the dimension of Vn will be calculated for each n e {0,..., iV —

1}. It corresponds to the number of irreducible subrepresentations of Çll[Xjj] hav¬

ing character Xn

1.4. Abelian Varieties for Appell-Lauricella Hypergeometric Series 73

1.4.8 Basis of Regular Differential Forms on XN

In this section, we intend to calculate a basis of the C-vector space Q1[Xn] of

regular differential 1-forms on X^. In view of the decomposition (1.21), §1.4.7,

it is sufficient to find a basis of Vn for each n e {0, ...,N — 1}. Once this being

done, we will calculate dime Vn by counting the basis elements and also the sum

dim Vn + dim V^-n» when (n, N) = 1.

Let's first make use of a results that goes back to Abel and Riemann and that is

stated in Satz 1, §9.3 of [BK] in the following way.

Proposition 1.9. The nonvanishing holomorphic differential 1-forms on the Rie¬

mann surface C, which is the desingularization of an irreducible algebraic planecurve C with affine equation fix, y) = 0, where the coordinates are chosen in

such a way that -^ is not identically zero, are given by [the pull-backs under the

desingularization map of]

<£(x, y)dx

df, ^~'

where <£>(x,y) = 0 is the equation of an adjoint curve to C ofdegree {degf) — 3.

We do not want to introduce what an adjoint curve is, but this proposition allows us

to choose a basis of regular differential 1 -forms on Xm among the regular pull-backsunder jt of the differential forms

<3?(x, y)dx(1.22)

yN-l

on Cyy, where 0(x, y) e C[x, y].

Since, for a point (x,y) £ CV, each potence y^, k e N, can be replaced by a

polynomial expression in x, we can suppose that

$(x, y) = <DoW + &i(x)y + ... + %_iW/_1.

That is

<3?(x,y)dx o(x)dx $>i(x)dx <bu-2(x)dx

yN-\=

yN-l+

N-2+ - + + ®N-l(.x)dx.


Hence, if the regular pull-backs of the differential forms (1.22) generates Qv[Xjq~\,so do the regular pull-backs of the differential forms

fy(x)dx

where n e {0, ...,N — 1} and \I>(jc) e C[x]. Further, the polynomials ^(x) will be

replaced by polynomials that are fitter to reflect the topology of Cm (resp. Xm) and

that also generate the ring C[x]. Namely, polynomials of the form

r

Y[(x-K)a' eC[x], a, eZ.

This discussion may be summarized by saying that the regular pull-backs under re

of the following differential forms on Cm

M*. y) = ; .

y11

with a, eZ and« e {0, ...,iV - 1}, generate £21[Xm].

Regularity Conditions for jt*con

Let's now fix n in {0, ...,N — 1}. We are looking for conditions on aç,, ..., a, for

n*con to be regular on Xm-

• On the dense subset U := Cm — {oo, (Xlt 0); j = 0,..., r} of Cm, the differ¬

ential form o)n is obviously regular, because (jc, y) h» x and (x, y) h-> -^ are

regular functions on U. Since the desingularization map tc : Xm -> Cm is a

morphism, the pull-back ir^con is regular on tv_1 (£/).

• AZ?ove(i : À, : O)Let 7 e {0, ..., r} and 2; e -X"iv be such that 7r{Qj) = (Xj, 0). As calculated

in (1.17) (last subparagraph of §1.4.5), the composition of it with the local

parametrization <pj of Xm at Qj reads (we use here the affine coordinates on

CN)


where gj (x) := Yliî (x ~ î)A' an] + kj ) ^ 0. By definition, we have

((7tj o C57)*(ü)B))(j) = ù)((7tj o Ç07)(j)) o ds(7tj o I?,)

andby Remark 10, §1.3.10, d^ ofp )(S)Xods(7tjO(pj) = n!°^' 1

(s)dss. Hence

((TTj o^)+

(wn))(j)

N

(N,Aj) n;=0(^(WA/) +kJ-klr'siNAJ) ^vg^j^v+^r^sAT

,^-"^

= C(5)//(WA/)+(A'A/) lds,

N

where C(j) - (äÊU;(*("V + *; " *0fl'£,(*(" V + *;)"* does not

take the value zero on Uj and is regular (because g} (s(N A<] + k; ) ÔonUj).Remark that this amounts to replacing x and y by their expressions in s and dx

by (A/^ ^(N Ai] ds in con(x,y). Since ^ is an analytic isomorphism, it is an

algebraic morphism and so is its inverse. This has the consequence that 7t*a)nis regular at Qj = (p} (0) exactely when (itj o^j)'<(û)„) is regular at 0 (because

it*(un = ((py1y((7rJo(pJ)*(ù)n)) and (tt7 o(pj)*(ù>n) = (<Pj)*(7t *&„)). Hence,

we have

nAj + (N,Aj)ic con is regular at Qj -<==>- a} > — - 1.

Note that this condition ensures the regularity of 7r *&>„ at each 7r-preimage of

(kj,0).

• Above InfinityFirst of all, we have to write the differential form con in projective coordinates.

Setting x := a and y := ^, we get

dx = —dx\ ^dxQ

and

r

cùnixQ, xi, x2) = x~hx'q" k=°ak ]~[(xi - KxçsY1 (x0dxi - xidxo).

i=0


Case h N -J2rk=oAk >0The composition of tv with the local parametrization ^coi of X^ at each

preimage Q of (0 : 1 : 0) is given by (cf (1.18) §1.4.5)

N-T,Ak

s , > (S(N.N-ZAk) • 1 - j(*,J'-EAt)|;(j(iV.»-E1i))I),

where h(x) = Wi=o^ - h*)Al and h(s^-N-Â^) ^ 0 for s e Uœ.

Noting that x\ = 1 => dx\ = 0 and inserting the expressions for

xq,x\,X2, dxQ into that of a>n, we get

((7r0oio^0Oi)*û;n)(j) = C(^ <-E**> ds,

where C is regular on U^ and C(,s) ^ 0 for 5 e Uoo- Therefore

7t*con is regular at Q

Owe 2: N - J^k=o Ak < 0

The composition of it with the local parametrization <poo2 of Xjv at each

preimage Q of (0 : 0 : 1) is given, for s e C/oo, by (cf (1.19) §1.4.5)

j 1 > (S(N.-N+EAk) ; ^(W.-Af+EAt) ; fr(s (W.-/V+E At) ) 77) .

Replacing xo, xi, X2, dxç>, dx\ by their expressions in s, we get

<iT.Ak-NY.aj-N -,

((7Tco2 o ôo2)*W„)(j) = C(J)J C-"+^*' V

where C is regular on L^ and C(s) ^ 0 for s e C/qo- Thus, we get

(7r*£ü„)(j) is regular at Q

j=0


Summary 2. Since (N, —N + Ylkô^k) = (N, N — JÂk), we can summarize

these conditions by saying that the pull-back under 7t : X^ -> C/v of the differential

form

con(x,y) =

on Ca? is regular on Xpj if and only if

YXl=Q{x-ki)a'dx

y-'' n- <nT!k=oM-(N,N-j:,kÂk) 1

2^=0 ai- N

l

fl; > —^—~ - 1, Vy {0, ...,r).

(1.23)

These conditions will be refered to as the regularity conditions for Jt*con.

Remark 36. We would like now to show that the pull-back n*con of a differential

form con(x, y) = y~n Yïi=o(x ~ îY'dx belongs to the isotypical component Vn of

character Xn, if it satifies the above conditions. If it is the case, Tt*con e ^[Zjy]and it remains to study the action of n^ on Tt*con, for a fixed n e {0, ...,

N — 1}.Let £ e /x^, then

£ • n*ojn

Gp?ojr)*<un by (1.20), §1.4.7.

Now, for P e Xn, we have

((<Pf O Tt)*COn))(P) = Cûn(SsPç O 7T)(P)) O dp(çof O TT)

= o)n{(pç{iï{P))) o d„(P)<pç o JP7r

= £nû;n(7r(P))odpjr

= r(?r*û>n)(P).

Hence, for every f e /x/y, we have

£ • (7t*COn) = Xn(Ç)x*(On.

This shows that Jt*co„ G V,


Dimension of Vn

Let n be fixed in {0, ...,N — 1}. In order to determine the dimension of Vn, we

will count the number of elements in a maximal family of linearly indépendantdifferential forms of the form y~n nC=o(x "~ î)aidx, where n, üq, ..., ar e Z and

satisfy the regularity conditions (1.23). Since uq, ..., ar are integers and according

to the regularity conditions, the maximal possible value (XX=o a*')max °f Xw=o ai

and the minimal possible value (a/)min of ay, j e {0, ..., r}, are given by

lZî=Oa^max — I ]vj ancl

/ n |"i nAj+(N.Aj)~\ •

^ rn -,

(aj)mm =-

[1 Sv J ' J e {°> •-' r^'

where [x] denotes the integral part of x.

Terminology 3. Let xel, then jc admits a unique decomposition as

x — [x] + {x},

where [x] e Z and (x) e [0, 1) are respectivelly called the integral part and the

fractional part of x.

Write further (£i=0a«)min := £-=0(ai)min and£ := Cl]i-ofl«)max- (E[=o^')min-

If I > 0, there is at least one solution. Write ojmin for the solution where each at is

minimal. Then Vn =< xkcüm{n >k=o,...,i and dim Vn =1 + 1. Indeed, one verifies

that each possible value for X^=o ai brings exactely one element in the maximal

family of linearly indépendant differential forms. For instance, if 3/ e {0, ..., r}

such that (flj)min + 1 and (^)-_0 ßj)min + 1 satisfy the regularity conditions, then

y-'\x - Xj)^^+1 Y\(x - Àf)(fl')min = XûJmin - kjCOnù 6< û)min, Xû;min > .

Note that this is indépendant of j and conclude by induction on I.

Theorem 1.10. Let X^ be the curve defined in Theorem 1.8, §1.4.6, and recall (cf

§1.4.7) that the space Q, [X/v] of regular differential 1-forms on Xn furnishes a

linear representation of'/x#. Then, for n 6 {0, ...,N — 1}, the isotypical component

Vn of character Xn has dimension

dim Vn =àn ifdn > 0

0 otherwise,


where

cln .—

nj:Ak-(N,N-j:Ak)N

+Ei=0

nAi + (N,Aiï

N

Proof. Use [x — 1] = [x] — 1 to show that dn =1+1 and apply the above

reasoning. D

Remark 37. If dim Vn = 0, then Vn does not appears in the canonical decomposi¬tion (1.21), §1.4.7, of GHXnI

Remark 38. Since by Definition 1.45, we have g[X^] = dime(£"21 [X^]), Theorem

1.8 and Theorem 1.10 together imply the following relation

1 +1-(rN-2

-(N, N -

k=0 7=0

Aj)) =- E7ie{0, ..,JV-

dn>0

Cln

-1}

Dimension of Vn, (n, N) = 1

Here will be used the conditions N \ Aq, ..., Ar, Xjj=0 A&. The goal here is to

transform the formula for dim V„ (cf Theorem 1.10) in the case where (n, N) = 1

into a more treatable form. Some preparatory lemmata are given in order to prove

Theorems 1.15 and 1.16.

Lemma 1.11. Let x e E, I e Z, N, A e N andn e {0, ...,N - 1}. Then we have

J. [x+i] = [x] + £,

2. [x] = x — (x),

3. {x+l) = {x}.

4. Ifx £ Z, then {-x) = 1 - (x)

,nA-(N,A)\ mA\ (N,A)5.IfN\Aand(n,N) =l,thenx N ,

_

x ^ ,-

N

6. IfN \ A and (n, N) = 1, then [-nA+Â)] = -[nA~(NN'A)] - 1.


Proof. The first four points follow directely from the definitions. For point 5., write

N' := (j^j, A' := ^y and nA' = kN' + r, withk e % andr g {1,.... N'-l}.

Note that rÔ, because N \nA. Then

nA' r nA' — 1 r — 1= k H and = £ H .

N' N' N' N'

Since r - 1 G {0, ...,N' - 2} and & g Z, we have

nA' r nA' — 1 r 1{ }

= — and { )=

.vN'

'N' N'

'N' N'

This implies C1^1} = (^) - ^ or equivalently

nA - (N, A) nA (N, A)( —L~1) = (—) -

NN N N

6. With the same notations and hypotheses as above, we have ["^T1] = k. Now,

-nA' - 1 r + l r + 1[ ] = [-k —] = -fc + [ —] = -fc - 1,

iV' iV' N'

because -^ G [-1, 0). Hence, t-^1^] = -V1^1] ~ 1 or equivalently

nA + (N, A) nA-(N,A)[ ]

= -[v

;] - 1.iV

J LN

J

D

Lemma 1.12. Letn G {0,..., N — 1}, N, Aq, ..., Ar G N and suppose (n, N) = 1

and A?" j Aq, ..., A,, J^=o Ajfc. TTien, Vy G {0, ..., A^}, we have

1. [-"W^] = <^> -^

- 1 andN J

—

\ N ' N

? r» EUo ^t-(^.^-Et=0 Ail) I_

" EUo A^_

/"EUoAtx^- L # J — ^ \ N '•

Proof. The reference number refers to Lemma 1.11.

l.Fix j G {0, ..., #}, then

^»A.+y.A^j =_["A/-y.AJ)]_1 by5_

= _^/-^y)+(>»A/-y.AJ))_1 by2_

_

nA, (7/,Aj) ,;M7 (AU;) 1 ,

s-

w^

a^"^ \ w / w

y°yJ-


9 r«E^-(JV,JV-E^)i_

nJ]Ak-{N,N-Y.Ak)_

,«E Ak-(N,N~J] Ak). ,

9^- l at J —

iv \ n / uyz-

_

BEA*_

(jy.Jv-EAii)_

/«Eii\ , (N,N-Y:Ak) h,

—

N N \ A7 ' "•"iV u->J-

_

«E4—

N

'E At,A7 '

D

Proposition 1.13. 7jf (n, iV) = 1 and iV { Ao, ..., Ar, Eîfc=o A^, ï/î<ot the integer dn

defined in Theorem 1.10 is equal to

dn =nErk=oAk

NEn

A;(-AT'

i=0N

Proof.

j_ \nY.Ak-{N^-Y.Ak)

an —

[ N1- rcA^ + ÇAT.A,)

A7+ E-=o

= ["

] + r + l+El~

— /"E^\ i y^/nAM

7iA,+(A/,A,)-|

N J

A7 Af

The second and third equalities are respectively obtained by applying 1. Lemma

1.11 and Lemma 1.12. D

Under the hypotheses (n, N) = 1 and N \ Aq, ..., A,-, Yllô Ak, we still can get a

better result on dim Vn. For this, we will use the following lemma.

Lemma 1.14. Let xq, ..., xr be real numbers. Then we have

r r

z=0 t=0

Proof First remark that

because of ^[xj] <s Z applied to 3. Lemma 1.11. Then we have

:: c,


by the above and by definition. Pay attention to the fact that c is an integer. Since

^2{*i) > 0 and —(%2xi) e (~ 1> 0]> the integer c cannot be negative, because

— 1 cannot be reached. Moreover, c < r, because J2(xt) < r + 1. Hence c e

{0, ...,/}. D

Theorem 1.15. Lef f/ie notations be as in Theorem 1.10 and suppose (n, A7") = 1

and N \ Aq, ..., Ar, Efc=o A^. 77zgn we /zave

«Et-n At-

x-î nA,dhnVn = -(

^*-°k) + £(_I)

iVi=0

N

Proof. By Proposition 1.13, we have

dn = -

,»ELoA*,A7

EnA,i=0

iV

and by Lemma 1.14, we know that dn e {0, ..., r}. Hence, by Theorem 1.10,

dim Vn = dn. D

dim Vn + dim VN-n, (n, N) = 1

Theorem 1.16. Ler f/ze notations be as in Theorem 1.10 and suppose (n, N)and N ] Aq, ..., Ar, X!L=o A-k- Then we have

= 1

dim Vn + dim Vn-u = r.

Proof.

dim Vn-h,{N-n)Y,Ak,\ N

, y^; , (N-n)Al+Li=0» N

-(EAk-^frL) + UA TlA,N

«EÛ

= -1 +

(-(;

N

\ n >

Y.Ai

) + E( nAj_<"

N '

+ r + l-U1W)

N+ E(ZF»

= r — dim Vn

(by Theorem 1.15)

(by 3. Lemma 1.11)

(by 4. Lemma 1.11)

(by Theorem 1.15).

D


1.4.9 New Forms

We are now approaching our goal of constructing an abelian variety on which

[ 7t*a>\ lives as a period. We could have taken the Jacobian variety of Xm, but

its dimension (equals to dim^LY^] = g[X^]) would have depended not only on

N but also on Aq, ..., Ar (cf Theorem 1.8). That is the reason why we will restrict

ourselves to a abelian subvariety of Jcic(Xm), whose dimension depends on N only.

In order to define this subvariety, we will select regular differential forms on Xm,

which "do not come from under" and are therefore called "new". This will be made

more precise.

First of all, let's work at the level of the singular curve Cm, because it is here possi¬ble to work with explicite expressions for the differential forms, in coordinates that

we choose to be affine.

Let d e N. If d\N, then we have a well-defined morphism

fd ' CN —> Cd

(x,y) i— (x,yd)

OO 1—> oo.

Let (u, v) e Cd be an affine point, then

-i --1

fd {(«>")} = {(M>uo).(H,fiVUo), ..., (K.fjv u0)},d d

where vq is any fixed -r-th root of v and Çn := e~R~.We see that there is an open

Cld

dense subset of Cci of points having ^ preimages. The other points have exactelyone preimage. (i/^ is actually a ramified topological covering.) The set of preimagesof a point P e Cd is called the fiber over P with respect to if/d-

Remember that the group ßM of iV-th roots of unity acts on Cm (cf §1.4.7) and

remark that the subgroup Id := (ÇN), Cm '= e N,°f index d in jxm acts transitively

on each fiber by permutation.

In the same way we had defined the induced action of //,# on S11[Xm] (cf §1.4.7),

we can deduce from the actionfiMxCm—

Cm


of iijq on Cm an action of /x# on the set <$>[Cn] of differential forms on Cn by

setting

HN x <P[CN] —> ^[Cw]

(£, w) i—> (p*co.

Now, suppose that you have a differential form r\ on C^. It is clear that its pull-back

ifjT) on Cn is invariant under the action of the subgroup Id, because Id preserves

the fibers.

The converse is more subtle. Let co G ^[Cn] be invariant under the action of Id-

Does co define a differential form (i(fd)*cD on Q;? The answer to this question is

positive, because Id acts transitively on each fiber. Hence, for Q e Cd, we can

define ((iAt/)*^0(o) to be the unique linear form on 9cd,Q such that, for P e Cn

with i/fd(P) — Q, {{ird)^){Q) ° dpfd = (o(P). This is well-defined, because

Vi3' G Qv with i//,/(P0 = Q, 3£ g 7rf such that <^(P;) = P and then

tü(P) = œ(n(P'))odP,n = (<p*çû>){P') = co(P'),

by invariance of co under 7f/. Remark further that ^((V^)*^) = <*> Indeed, let

P e Cn, then

tdMd)*<o){P) = Wd)*co)Wd{P)) o dpfa = <w(P).

The so-defined differential form (\Jfd)*oj G <£>[0] is called the push-forward of co

with respect to i/^/.

For a differential form &> on Cn and d| TV, we have shown

co is fixed under the action of Id on $>[Cn]

•<=> 3?7 G <£>[Cd] such that i/^?? = co.

Let's now consider the differential form con(x, v) = y~n ]~C=o(x — 'kiY'dx, ai e 7L,

on Cn Under which condition on n is con fixed by the action of Id ? Well,

Vf Zi, ÇOçfun = con «=> Vf /,/, Ç"(On = 0)n

<=» Vk G {0, ..., f - 1}, {çff)nCOn = 0)n

^=^ leZs.t. n = ^f.


The differential forms which satisfy this for a d dividing N and different from N are

the ones we want to get rid of, because "they come from under". This is equivalentto the fact that (N,n) ^ 1. Indeed, if the above equivalent conditions hold, j is

^ 1 and divides N and n. Conversely, suppose that (N,n) ^ 1, then con is fixed

under the action of I_n_ . Hence, we define(N.n)

Definition 1.73. A differential form con (resp. 7t*con) on Cm (resp. Xn) such that

(n, N) — 1 and the linear combinations of such differential forms are said to be

new. The vector subspace of ^[Xn] consisting of all new forms on Xn which are

holomorphic is

&> [Xpf]new := (^ Vn,

where the sum is taken over the n e {0, ...,N — 1} such that dim Vn > 0 and

(n,N) = l.

1.4.10 New Jacobian

In the previous paragraph, we have define a vector subspace Q1 [X^Jnew of ß1 [Xn],

consisting of the regular differential forms on Xn, which are not the pull-back of a

differential form on Xj for any d dividing N.

According to Remark 25, §1.3.16, recall that the Jacobian variety Jüc(Xn) of Xnis the following abelian variety

&[XNT/i(Hi(XN{C),T)).

By Remark 24, §1.3.15, the vector subspace ^[X^new defines an abelian sub-

variety of Jcic(Xn). It will be called the New Jacobian of Xn and denoted byJacnew (Xn) .

Furthermore, Jacnew{XN) has a complement C (as abelian variety) such that the

abelian variety Jo,c(Xn) is isogenous to

Jacnew(XN) © C

and dim(Jac(Xjs[)) = dim(7acneiu(X^)) + dimC. The dimension of the abelian

variety Jacnew(Xj^) equals that of the C-vector space Çll[XN~\new Because

à\m(Q}[XN]new) = ^2 dim(K) = - ]P (dim V„ + dim Viv-,0

(n,N)=l (n,N)=l0<n<N 0<n<N


and because of the general assumptions N \ Aq, ..., Ar, J2'k=o ^k, we have dim Vn+

dim Vn-u — r by Theorem 1.16, §1.4.8. Hence,

r<p(N)dimJacnew(XN) = —-—

where <p(N) \= J2 1 is the Euler function.

(n,JV)=l0</i<JV

1.5 Abelian Varieties for Gauss' Hypergeometric Se¬

ries

1.5.1 Motivation

Consider a Gauss' hypergeometric series F {a, b, c; z), where a,b,c e <Q, c =£

0, —1, —2, ... andz e C. If \z\ < 1, the series converges and if Re(c) > Re{b) > 0,

we can write

t (a, b, c\ z) = ,

where P(z) = f0 xb~l{\ -x)c~b~l{\ -zx)~adx. As explained in Section 1.2, our

goal is to find an abelian variety on which a given algebraic multiple of P(z) is a

period. In this section, we treat the case zÔ, 1. Let JV be the smallest common

denominator of a, b, c and set

A:=N(l-b), B:=N(l+b-c), C := Na.

Then A, B,C eZ, N e N and the affine equation

yN = xA(l-x)B(l-zx)c

defines an algebraic projective curve over C denoted by C(N, z). With this notation,

we have

Cl dx

Joy

'

/oy

7.5. Abelian Varietiesfor Gauss' Hypergeometric Series 87

If a, b, c — b <£7h, this integral can be replaced, up to a complex algebraic factor,

by an integral f ^f, where y is a closed path on C(N, z) whose projection in C

is a Pochhammer loop around 0 and 1. We have gained that this integral converges

uniformly for every z e C — M+ without condition on Re{b), Re(c).

Let X(N, z) be the desingularization of C(N, z), 7tz : X(n, z) -» C{N, z) the

desingularization map and œ := ^f. If n*co is regular onX(N, z) and if S represents

a class C in H\ (X(N, z)(C), Z) such that (7rz)*<5 = y, then the integral

/ 7T*(0 = I OJ

Js~

J(k)«s

is independent of the choice of the représentant of C (cf § 1.3.13) and is a period on

Jac(X(N, z)). We will actually define an abelian subvariety on which fs 7t*co is

still a period.

Throughout this section, we will assume that

N\A, B, C, A + B + C,

what implies

a, b, c — b, c — a $ TL.

In prevision of Section 1.6, we will also assume

c^TL.

The excluded cases correspond to hypergeometric series, whose transcendental na¬

ture is well understood (cf [Wo88] §3). Moreover, we will suppose the curve to be

irreducible, i.e.

(N,A,B,C) = l.

In the next paragraph, we show that X(N, z) is isomorphic to a given algebraiccurve of the family considered in Section 1.4. This allows us to use in §1.5.3 our

results of Section 1.4 to calculate the genus of X(N, z), a basis of regular differen¬

tial 1-forms and to define the New Jacobian of X(N, z) on which fs 7t*co lives as a

period, if Tt*a> is regular onX(N, z). This will respectively correct and prove some

assertions of Wolfart, [Wo88].


1.5.2 The Isomorphism

As announced in the Motivation, the aim is here to find an isomorphism (of alge¬braic curves) between the projective algebraic curve X(N, z) and the desingulariza-tion of the projective curve with affme equation

(=0

where Aq, A\, A2 Z and N e N. Let's first work at the level of the singularcurves. Recall that C(N, z) is affinely defined by

yN = xA(\-x)B(l-zxf,

where zeC-(0, 1}, A,B,C gZ, N eN, N ^ I, A + B + C. Define CN to be

the projective curve with affine equation

yN=xN-A-B-Cix_1)Bix_z)Ct

Xn its desingularization and n : X^ —>- Cm the desingularization map. Then the

following map is well-defined

k : C(N, z) — CN

(xq : x\ : X2) 1—> (x\ : xq : x2).

Indeed, using the notations of §1.4.3 and the coordinates (uq : u\ : u-£) on C/v, wehave N - J2l=o Ak = A and

CN : ul = uAu^~A~B~c(ui - «o)ß(wi - zu0f.

For the moment, let's accept negative exponents and simply write a factor with

negative exponent on the other side of the equality. With the same conventions, the

equation of C(N, z) in projective coordinates reads

C(N, z) : x% = x£-A~B-cxA(x0 - Xl)B(x0 - zxif.

Suppose that (xq : x\ : x-i) e CCA'', z) and write (uq : u\ : U2) '.— (x\ : xq : x^)then u^ ~ ul~

~~ ~~

uA(u\ — uq)b (u\ — zuq)c ,hence (uq : u\ : u^) e Cm and a:

is well-defined.


k is clearly a morphism of algebraic varieties, and because /c_1 = k, it is also

an isomorphism. This implies that the composition k o txz : X(N, z) —> Xm is a

birational morphism. Since X(N, z) is moreover nonsingular, it provides a model of

the desingularization of Cm and there exists a unique isomorphism ic from X(N,z)to Xm such that jr ok = k o nz (cf [Ful], Ch7§5, Thm3). Hence, fi![Xjv] and

ß^XCiV, z)) are in bijection under k*. Remembering that in §1.4.2, we had reduced

to positive exponents, we can here also suppose N—A — B — C, B, C > 0.

1.5.3 Genus and New Jacobian ofX(N, z)

Let X (N, z) with z ^ 0, 1 be the curve defined in the introduction of this section. In

Paragraph 1.5.2, we have seen that X(N, z) is isomorphic to the desingularizationXjv of the projective curve Cm with affine equation

yN=xN-A-B-C{x_l)B{x_z)C

and that we can suppose

N-A-B-C, B, C >0.

Moreover, the irreducibility of X(N, z) is equivalent to that of Xm, so that we are

allowed to apply the results of Section 1.4 to Xm-

Genus of X(N,z)

Because X(N, z) and Xm are isomorphic, we have

x(X(N,z)) = x(XN) and g[X(N, z)] = g[XN].

Applying Theorem 1.8, §1.4.8, we get

X(X(N, z)(C)) = -IN + (N, A) + (N, B) + (N, C) + (N, N-A-B-C)

and

g[X(N, Z)] = N+1- 1-[{N, A) + (N, B) + (N, C) + (N, N-A-B-C)]


Regular Differential Forms on XN

Write jt : X^ —> CV for the desingularization map and, for n e {0, ...,iV — 1}, let

xa°(x- l)ai(* ~z)a2dx

yll

define a (rational) differential form con on Cn< Then by the regularitiy conditions

(1.23), §1.4.8, 7t*con is regular on Xn if and only if the following four conditions

hold

n{N-A-B~C)+{N.N-A-B-C)_ -,a0 >

a\ >

a2 >

aç, + a\ + a% <

nB+(N B)_

,

N '

(1.24)nC+(N.C)

_

,

N *>

n(N-A)+(N,A)_ -,

Nl-

If, for n e {0, ...,N — 1}, Vn denotes the isotypical component of Q1[Xn) of

character Xn, then, by Theorem 1.10, §1.4.8, we have

dim Vn —dn if dn > 0

0 otherwise,

where dn is equal to

.

MA + B + C)-(N,N-A-B-C)^ri

nA + (N,A)^dn = [— 1 + [1 1

N N

,

nB + (N, B) nC + (N, C)+ [1

V

;] + [lV

;] (1.25)

(in order to get dn in this form use that, if k e Z, then [x + &] = [x] + £).

Remark 39. If ??n denotes the (rational) differential form y~nxb°(l—x)hl (l—zx)b2dxon C(N, z), the conditions (1.24) can be used to determine when Tt*r]n is holo-

morphic on X(N,z). Indeed, K*rjn is holomorphic on X(N,z) exactely when

7T*(k~1)*î],1 is holomorphic on Xp/, that is, when the following four conditions

hold

bo > ^Â)_1.

> nB+{N,B)_ 1

*2 > ^±^-1èo + è1+è2 < "(A+i»+c)-(jy,jy-A-j-c) _ L


This corrects the assertion of Wolfart ([W088] §4) on the holomorphy conditions

for differential 1-forms, while his assertion on the dimension of the isotypical com¬

ponents Vn for n G {0,..., N — 1} is corrected by (1.25).

Supposing now that (n, N) = 1 and using Theorem 1.15, §1.4.8, we get

dim vn = -<^) + r(N-Y~C)) + (f) + <f)

and, using that, Vx el\Z, (—x) = 1 — (x),

nA nB nC n(A + B + C)dim Vn = (—) + (—) + (—> - {— —).

And finally, applying Theorem 1.16, §1.4.8, we have

dim Vn + dim Vn-h — 2

in the case where (n, N) = 1.

New Jacobian of X(N, z)

In paragraph 1.4.9, we have selected so-called new forms on X^. In §1.4.10, it

has been shown that the vector space Q}[X^]new generated by these new forms has

dimension j<p(N) (here r — 2) and the abelian subvariety Jacnew{X^) of JacCX^)

generated by Ç21 [X^]new has been called the new Jacobian of X^. If we set

!^[X(iV, Z)]new := ~K*{£ll[XN]new),

we have

dim(&[X(N,z)]new)=(p(N)

because ic is an isomorphism.

Definition 1.74. The abelian subvariety of Jac(X{N, z)) generated by the vector

subspace Sll[X{N, z)]new of £ll[X(N, z)] will be called the New Jacobian of

X(N, z). It has dimension <p(N) and will be denoted by Jacnew(X(N, z)).


Conclusion

Recall that the following diagramm commutes (cf §1.5.2)

X(N,z) -^-* XN

C(N,z) > CN,K

where it, nz are morphisms and k, k isomorphisms. Write co\ for the differential

form (k~1)*oo on Cjq, where co is the differential form c-y on C(N, z) (hence k*cù\ =

co). One verifies that tt*co\ is a new form on Xpj-

Remember that our goal was to construct an abelian variety on which f co lives

as a period, if y is a given cycle on C{N, z). Let jt*y be a (not unique) cycle on

X(N, z) satisfying (7rz)*7r*y = y.

If tx*(jO\ is regular on X^, our task is achieved. Indeed, in this case, i<*(7t*coi) is

regular on X (AT, z) and is an element of Q} [X (AT, z)]new Remark that

K*(7T*00l) — (TT O k)*COi = (K O 7tz)*OJl — 7t*(lC*ù)l) — 7t*Cü.

Hence, the integral

/ ic*(7t*COl) = I TT* 00 = I CO

is a period on the abelian variety Jacnew(X(N, z)) of dimension <p(N).

1.6 Abelian Varieties for the Beta Function

1.6.1 Motivation

We would like here to construct an abelian variety on which an algebraic multiple of

the integral in the denominator of the integral representation of the hypergeometricseries F(a,b,c;z) lives as a period. The procedure will be analogous as that in

Section 1.5. That is, the integral P(0) will be written as / ^, where (x, y) lies on a

1.6. Abelian Varieties for the Beta Function 93

certain curve which is isomorphic to a curve of a subfamily of the family for which

we have had results in Section 1.4.

First recall that the integral representation of F {a, b, c; z) reads

F{a, b, c; z)p(oy

where !P(z) = f0 xb 1(1 — x)cb 1(l — zx) adx. In Section 1.5, we had treated

the case of 3>(z) with zÔ, 1. Let's now treat the case of ^(0). First remark that

p(0) = B(b,c-b),

where B(a, ß) = fQ xa~l{\ — x)@~ldx is the classical Beta function, which con¬

verges if Re (a) > 0 and Re(ß) > 0. Set M e N for the smallest common denomi¬

nator of b and c. Set further

P:=M(l-b), Q :=M(l+b-c)

and define X(M, 0) to be the desingularization of the projective curve C(M, 0)defined over C by the affine equation

yM=xP{\-x)Q,

where P, Q eZ. Write œ for the differential form — on C(M, 0), then we have

3>(Q) = f co.

Jo

If b, c — b £ 7j and if y is the lifting on C(M, 0) of a Pochhammer loop in C

around 0 and 1, then the integral f co is equal to an algebraic multiple of fQ co and

converges without restriction on Reib), Re(c). If S is a cycle on X(M, 0) suchthat

(7To)*<5 = Y and tio : X(M, 0) -> C(M, 0) the desingularization map, then

/ JTqCO = I CO.

Consequently, it is for us sufficient to construct an abelian variety on which fs jVqColives as a period.

Before going further, let's recall the hypotheses (cf §1.5.1). They read

a, b, c e Q — Z and a — c, b — c $ Z

and implyM\P, Q, P + Q.

In particular, M^l, P+ Q.


1.6.2 The Isomorphism

Consider the algebraic projective curve Cm affinely defined over C by

yM=xM-P-Q(x_l)Qj

where MeN, P, Q Z. Then, the projective equation of Cm reads

%2 = Xq X^ v^l -Ô) >

with the convention that a factor with negative exponent should be written on the

other side of the equality. With the same convention, the projective equation of

CiM, 0) is

XM =xM-P-QrPfrn_^Q0 *xx (x0 -xi)^,

and we can define a map

k : C(M, 0) —> CM

(xq : x\ : x2) i—> (xi : xo : xi).

This map is an isomorphism, because it is a morphism equal to its inverse.

Write now % : Xm —>- Cm for the desingularization of Cm- Since k o jïq and it are

birational maps from a nonsingular projective curve to Cm, there exists a uniqueisomorphism k : X(M, 0) —>- Xm such that the following diagramm commutes

X(M,0) ^-> XM

C(M,0) > CM.K

In particular, this shows that X(M, 0) and Xm have same Euler characteristic and

genus, and that ^*^1[XM] = Sll[X{M, 0)]. We will here also suppose X(M, 0)and Xm to be irreducible, i.e.

(M,P,Q) = l.

Thanks to §1.4.2, wecan also suppose thatM—P—ß, Q > 0 and apply our results

of Section 1.4.

1.6. Abelian Varietiesfor the Beta Function 95

1.6.3 Genus and New Jacobian of X (M, 0)

Genus of X(M,0)

Applying Theorem 1.8, §1.4.8, we get

X(X(M, 0)(<Q) = -M+ (M, P) + (M, Q) + (M, M-P-Q)

and

g[X(M, 0)] = 1 + i[M - (M, P) - (M, ß) - (M, M-P- ß)]

Regular Differential Forms on Xm

For n e {0,..., M — 1}, let con be the (rational) differential form on Cm defined by

xa°(x- ly^dx

yII

By the regularity conditions (1.23), §1.4.8, 7t*(ûn is regular on Xm if and only if

n(M-P-ß)+(M,M-f-i2)_

i

"°— M i;

fll > »6+j[M,ß) _ 1; (1.26)

Remark 40. For n e {0, ...,M — 1}, let r\n be the (rational) differential 1-form

y-nxbo(l - x)bidx on C(M, 0). Then jt*rjn is regular on X(Af, 0) if and only if

7t*(K~l)*rin is regular on Xm- By the conditions (1.26), this is the case exactelywhen the following three conditions hold.

h > ^F1-!,u, >

nQ+{M,Q)_

-,

Ul- M i'

bQ + bl <»C+Q)-(M.M-P-Q)

_ LM


Forn e {0, ..., M—l}, let Vn be the isotypical component of ^[Xm] with character

Xn- Then Theorem 1.10, §1.4.8, implies

dim Vn =

where

dn if 4 > 0

0 otherwise,

,_

fn(M-P)~(M,P)-] r1 n(M-P~Q)+{M,M-P~Q)1, n «6+(M,0-,

"« — L M J -I" U M J I" Li M J

_

rn(f+g)-(M,M-P-g)-, , n /iP+(M.F)-| , F1 nQ+jM,Q)1— L M J t[l M J-t-L-l M ]-

Suppose now that («, iV) = 1, then it follows from Theorem 1.15, §1.4.8, that

— \ m I I- \ M ; \ M /

and from Theorem 1.16, §1.4.8, that

dim Vn + dim V/v_n = 1

New Jacobian of X(M, 0)

According to §1.4.10, the vector space Q}[XM~\new of new differential forms on

Xm has dimension ^r^-- êt &l[X(M, 0)]new := /c*(ß1[X^]raew). Because k is

an isomorphism, dim(^*(Q1 [XMlnew)) = dimiti1 [XM]new) and

dim(^X(M,0)]new)==-^Definition 1.75. The abelian subvariety of Jac(X(M, 0)) which is generated byQ}[X{M, 0)]new will be called the New Jacobian of X(M,0) and denoted byJacnew(X(M, 0)). It has dimension ^.

Conclusion

Write co i for the differential 1 -form (k )*co on Cm ,

where <ï> is the differential form

Yon C(M, 0). Then co\ is a new form on Cm and, if 7t*co\ is regular on Xm, then

1.7. Degenerated Fibers 97

k*(ix*(o\) e £ll[X(M, 0)]new Remark that, in this case, jTqCo e Q>l\X(M, 0)]new,

because k*(jt*coi) = 7rQ(K*œi) — tTqCO. If 5 is a cycle on X(M,0) such that

(tto)*8 = y, where y is a cycle on C(M, 0) whose projection in C is a double

contour loop around 0 and 1, then the integral

7Tq CO = / Co

<y

<P(M)is a period on theq-1

-dimensional abelian variety Jacnew(X(M, 0)).

1.7 Degenerated Fibers

In this section, we would like to enlarge the family of curves treated in Section 1.4

by considering the parameters Xq, ...,Xrto lie in Pj, and allowing them to be equal.The nondegenerated curves of this family will however be those of the family of

Section 1.4, their genus and the dimension of their New Jacobian being then known.

We here intend to calculate the genus and the dimension of the New Jacobian of the

degenerated curves of the enlarged family. This is actually again an application of

Section 1.4 and will be done in § 1.7.2 after we have defined the enlarged family and

introduced new notations in § 1.7.1.

1.7.1 New Notations and Definitions

For r e N U {0}, N e N and u = (A0, A0; ...; K, Ar) e (Pj, x N)r+1 such that

Af { Ao, ..., A,, J2k=o Ak and (N, Ao,..., A, ) = 1, let us define CN(u) to be the

projective plane curve with affine equation

yN = Y\(x-Xiyi=0

and Xn(u) its desingularization.

If, Vz' = 0, ..., r, À, 7^ oo and, for all i, j e {0,..., r} with / ^ j, Xt ^ X}, then

Cn{u) = Cjq and X^{u) = X^ are the curves defined in §1.4.3 and the results of

Section 1.4 apply.


Remark 41. The projective equation of Cn(u) defines an algebraic projective va¬

riety T in Pj x (P^)r+1. For fixed r e N U {0} and N, A0, ..., Ar e N, the

curves C^{u) are isomorphic to the fibers of the restriction to !F of the projection

Pj x (Pj0;'+1 -> ÇP1c)r+ï. A fiber over (À0, ..., Xr) such that, Vi G {0, ..., r), Xt 7^

00, and, Vi, 7 e {0, ..., r} with i 7^ j, À, 7^ À; will be said to be nondegenerated.It is then isomorphic to a curve of the family considered in Section 1.4. The other

fibers are said to be degenerated. The same qualifyiers will also be used for the

curves themselves. In order to describe the degenerated curves, we introduce the

following definition.

Definition 1.76. Let m > 1 be the number of pairwise distinct entries in the (r +1)-

tuple (Xq, ..., A.,-) e (P^)r+1 and call it the multiplicity of u = (Xo, Aq\ ...; Xr, Ar).

Define the reduced form of u to be the element urec\ := (/xo, -Bo; ', ßm-i, Bm-i)

of (C x N)m_1 x (Pj, x N), where Vi g {0, ...,m - 1} : 3k g {0, ..., r} such that

fxl = Xjç and Bl = E Aj, where the sum is taken over all the j's in {0, ..., r) such

that Xj = jxt. Moreover, Vi, j e {0, ...,m — 1} with j 7^ i : /Xj 7^ /x(.

This definition is essentially only a new notation. Indeed, we have

m— \

CN{u) = CN(ured) : yN = Y[(x - i^)8'1=0

andXN(u) = XN(ured).

1.7.2 Dimension of the New Jacobian and Genus of some Fibers

Let Cn(u) be the curve defined in §1.7.1 and the notations and definitions be

the same. We will first treat the case /x;n_i 7^ 00. If CV(w) is irreducible (i.e.

(N, B0, ..., Bm-i) = 1) and if N ^ £L"o Bk (note that Eô Bk = ELo Aklthen the genus of X^ (u) is given by

1m— \ m— \

g(XN(u)) = 1 + -[(m - l)N -(N,N-J2 Bk) ~ J^(N, 5;)]k=0 j=0

(cf Theorem 1.8, § 1.4.6). If moreover N \ B0, ..., Bm-i, J2l=o B^ then the results

of Section 1.4 apply to the case \jlm-\ 7^ 00. In this case, we have

<hm{Jacnew(XN{u)))

1.7. Degenerated Fibers 99

(cf §1.4.10). Note that this dimension depends only on the number of pairwisedistinct factors in the equation of Cn(u).

For the case \xm-\ = oo, replace fim-\ by I1 in the projective equation of Cîured)and get

Casel:N-J2l=oBk>0:

x^"-1 = x"-^ Bk(xlZ0 - x0Zl)B»^ "ffoci - mxoY;=0

Case 2: N -J2T=d B* < 0:

m—2

i=0

x?Zom-'x0 N+EBk = (xlZ0 - xozi)B"'^ n (*i " /ô)5'

Set zo = 0 and get

N_Y^m-2d n

m—2

• Casel:0 = xQ^=° kz°m-1 Y\ {xx - ^xq)b'

• Case 2: 0 = x^-^f'""1 f] (*i ~ Mô)5'i=0

Since z\ j^ 0, the solutions in both cases are xq = 0 or 3z e {0, ...,m — 2} such that

xi = ^/xo, i.e.

Cn(u) = {(xo : xi : xi) e P^! xo = 0 or 3z e {0, ...,m — 2} s.t. xi = /x;-xo}

7?!—2

= {(0 : xi : x2) e Pj} U U {(*o : w*o : *2) e P^}.î=0

This shows that for ßm-i = oo, Cm (u) is isomorphic to m copies of P^ lying in P^,.These copies are the irreducible components of Cm (u) and the only singular pointsof Cn(u) appear to be the intersection points of at least two distinct copies. These

irreducible components are separated during the sequence of blow-ups and are iso¬

morphic to the connected components of the desingularization Xm(u) of Cm(u),because each of them is nonsingular (cf §1.3.7). Each connected component of

Xn(u) has genus 0 and 0 is also the dimension of the Jacobian variety of Xm(u).

Chapter 2

Identities for some Gauss'


2.1 Introduction

In their article [BW86], Beukers and Wolfart proved two identities involving a hy¬

pergeometric series, the /-function and the Dedekind ^-function. One of these

reads: there exists AeC such that, for all r in the neighbourhood of i given by|1 — J(z)"1\ < 1, we have

To get such an identity, Beukers and Wolfart made use of a result of Klein and

Fricke, which states that the periods on the "universal" elliptic curve satisfy a givenlinear differential equation of order two with regular singularities at 0, 1, oo onlyand of the theory of elliptic curves to get a local fundamental system to this differ¬

ential equation. On the other hand, they found two solutions involving hypergeo¬metric series and expressed them locally in terms of the fundamental system. This

way, they obtained an identity for each solution.

In the Theoretical Preliminaries, we recall some theoretical facts used throughoutthe chapter. In Section 2.3, we explain and extend the method of Beukers and

Wolfart. In Section 2.4, we give the list of the solutions involving hypergeometric

101

102 2. Identitiesfor some Gauss'Hypergeometric Series

series that we have calculated to the above linear differential equation. We also

explain how one can find them using Riemann's point of view and Kummer's idea.

In Section 2.5, we produce identities for our solutions, most of them being new.

Finally, Section 2.6 provides some applications of our identities. On one hand, some

identities allow us to show the algebraicity of the corresponding hypergeometricseries at the points of some infinite subset of Q. On the other hand, we are able to

calculate explicitely some new algebraic evaluations of F(^, ^, \\ z).

2.2 Theoretical Preliminaries

We recall here some definitions and facts on elliptic curves over C and some asso¬

ciated modular functions and forms.

2.2.1 Elliptic Curves over C

Let A be a lattice in C and pa the associated Weierstrass function on C

1v^ 1 1

Pa(z) = — + > [--^- -2].

Zz *—' (Z— (oY Cûz

û)6A-{0}V J

The series converges uniformaly on each compact set not containing any lattice

point and the only poles of Pa are double poles in each lattice point. Moreover Pa

is periodic with respect to A, what makes it an elliptic function. Differentiatingterm by term, one shows that pa and p'A satisfy the relation

(p'a)2 = 4Pa3 - <?2(A)pA - £3(A),

where g2(A) = 60 ^ ^ and g3(A) = 140 £ ^. This means that the

ûj6A-{0}°J

coeA-10}W

map z -> (Pa (z), p'a (z)) parametrizes points on the curve defined by the equation

y2 = 4x3-g2(A)x-g3(A). (2.1)

More precisely, writing £a(C) for the algebraic projective curve defined affinelyby (2.1) over C, we have the analytic (group) isomorphism

$A: C/A —> £A(C)

z mod A 1—> (1 : pA(z) : p'A(z)) forz ^ A

0 1— (0 : 0 : 1).

2.2. Theoretical Preliminaries 103

Each equation of the so-called Weierstrass' form y2 — 4x3 — ax — b with a, b e C

and a3 — 21b2 ^ 0 defines an algebraic projective curve Ea^ over C, which is non

singular and called elliptic. For each such curve, there exists a unique lattice Aaîn C, such that 3>a3 b

' ^M —> Ea,b is an analytic isomorphism. This lattice

could be referred to as the "corresponding" lattice to the curve Ea^. Its elements

are called periods of Ea^. In particular, a = g2(â.b) and b = g3(Aa,è) hold.

Let E be an elliptic curve over C given in Weierstrass' form by x, y coordinates. We

are now wishing to determine its corresponding lattice A. The analytic isomorphismOa : C/A —> E(C) induces an isomorphism of Z-modules

0A* : #i(£(C),Z) —> #i(C/A,Z)

[y] i—> [$A_1oy]

(cf §1.3.13). Moreover, Hi(C/A, Z) is isomorphic to A via the map

[5] i— f dz.

s

Hence, if G\, &2 form a Z-basis of H\ (E(C), Z), then _/",* edz, /0* e?

dz will form

a Z-basis of A. Now, for i = 1, 2, we have

/dx f dx f

7=

J ^=

J <h-

This implies

/dxf dx

hZ / —.

Ci e2

In fact, the inverse map of $a is given by

£(C) — C/Ap

P i—> (/7/z)(modA).o

Remark 42. The above map is a special case of the map given in Remark 27,

§1.3.16, which embeds a Riemann surface into its Jacobian variety. In particular,we see that each elliptic curve defined over C is isomorphic to its Jacobian variety.

104 2. Identitiesfor some Gauss'


2.2.2 Isomorphisms between Elliptic Curves

Let E : y2 — 4x3 — ax — b and E' : y' = Ax' — a'x' — b' be two elliptic curves

in Weierstrass' form over a field k of characteristic ^ 2, 3. They are isomorphic if

and only if there exists /x e k such that a' = fi~4a and b' = fi~6b. If k = C, we

get in terms of the corresponding lattices A for E and A' for E'

E' 3ß g C" such that A' = /xA.

More preciselly, we have the following commutative diagramm

C/A —!— C//xA

*M

-> £'(C)

*A

£(C)

Corresponding to these variable transformations, the following homogeneity prop¬

erties hold for any /x G C* and z G C

PnA(ßz) = m~2Pa(z) PßAr(ßz) = /x~3Pa(z)g2(M) = M"4g2(A) #3(M) = M~6g3(A).

2.2.3 Change of Basis of a Lattice

([Sei] VII §1) Let A be a lattice in C with Z-basis co\ and <x>2- Since co\ and

a>2 generate C over R, they are M-linearly independent. Hence Im(^) ^ 0 and

reordering a>\ and (02 if necessary, we can suppose Im(^-) > 0.~-C02'

Any new basis co\ , C02 of A has the form a>\ = aco\ + bco2, <*>2 = c^\ + da>2with y :— (acbd) g GLiÇfc). co\ , 0)2' is also positively oriented if and only if y has

determinant 1, that is y G ,5X2(Z). The action by linear transformations of SL2ÇZ)on H := {z G C; Im(z) > 0} is defined by setting, for y = (ac J) in SL2(Z) and

T GM,

ax + by x :=

.

ex + d


The quotient M/SL2CZ) for this action is in bijection with the set of lattices in C

defined up to homothety, hence also with the set of isomorphy classes of ellipticcurves over C.

Since / and —I are the only elements of SL%(J1) that act trivially on the whole of

H, the quotient group PSL2{Z) := SL2(Z)/{±I} acts freely on EL It will be called

the modular group (this denomination is sometimes used for SL2ÇZ) itself). A

transformation of H induced by the action of an element of the modular group will

be said to be modular.

Definition 2.1. Let F be a discrete subgroup of SL2ÇL). A fundamental domain

D for F in H is a subset of H such that every orbit of F has one représentant in D

and two elements of D are in the same orbite if and only if they lie on the boundaryofD.

A fundamental domain for PSL2ÇE) is the set of z e H such that

— < Reiz) < - and \z\ > 1.2-

2

Definition 2.2. The parabolic points of F in a fundamental domain D for F are

the real points together with 00 which lie in the closure of D for the topology of the

Riemann sphere.

00 is the sole parabolic point of 5X2(Z)/{±1}.

2.2.4 From a Lattice Function to a Modular one,

Definition of Modular Functions and Forms

([Sel] VII §§2.1-2) In §2.2.1, we have met two functions g2, £3 defined on the set of

lattices and satisfying some transformations properties under homothety of lattice.

More generally, a function F on the set SI of lattices in C with complex values is

said to be of weight 2k, k e Z, if for every Ae^ and ß e C*

F(M) = ß~2kF(A).

Write A = Zû>i + rLoj2 with ImQ^ > 0 and let F{a>\, co2) stand for F (A), then

the above property shows that œ%kF(coi, &>2) depends only on ^ and is invariant

106 2. Identitiesfor some Gauss' Hypergeometric Series

under any change of basis of A. Let / : H —> C be defined by

/(—)=CO%cF(a>i,C02),co2

it will then satisfy, for any (« \ ) e SL2(T) and z e H

/( -7) = {cr+d)2kf{x).ex + a

Noting that the transformation x m>- elltlx =: q takes H to the open disk D* := {q e

C; \q\ = 1, g t^ 0} with the origin removed and that, Vr e H, /(r + 1) = /(t),

we can consider / as a function / of q on D*.

Definition 2.3. Let & be an integer. A meromorphic function / : H —> C is said

to be modular of weight 2k if / is meromorphic at zero and if, for every x e C

and every y = (acbd) e SL2(Z),

f(yx) = (cx + d)2kf(x).

In this case, / has a Laurent development Xôo a«<?" at 0 anc*tne nature of / at oo

will per definition be that of / at 0.

Definition 2.4. A modular function is called a modular form, if it is holomorphicin H and at infinity. In this case, we set /(oo) := /(0). If moreover it takes the

value zero at infinity, it is called a cusp form.

2.2.5 Eisenstein Series

Remembering the functions g2, g3 we had met in §2.2.1, we consider here more

generally the so-called Eisenstein series

G*(A) = J2 —^coeA

for given integer k > 1 and lattice A in C, Y2' meaning summation over the non

zero elements. Observing that

Gjfc(AtA) = ^2kGk(A),


we can apply the procedure described in §2.2.4 to make it a function ofreH

Gk(cOl,(D2)= y^; -~T = —y: 2. , «i . vu-t—1 (mcoi + nco2)Zk oof ^_ (m^J- + ri)lk

Let's write again Gk for the corresponding function of r := — e M

v-' 1

Gk(r) = > 7 -—2^-*—' (inx + ?î)ZK

m,neZ

It can be shown that the Eisenstein series Gk is a modular form of weight 2k. This

authorizes us to write g2 and g3 as modular forms of weight 4 and 6 respectively as

follows

g2(r) = 60G2(r) and g3(r) = 140G3(r).

g2 and g3 are holomorphic in H and also at infinity. They have the following Lau¬

rent's development in q := elirir

g2(r) = 7r4[| + 320^ + ...]

Setting q = 0, we get the values at infinity

g3(T)=Jr6[§f-^fq+ ...}.

44

86

g2(°o) = -7T and g3(oo) = —tt.

2jn

For p denoting e 3, g2(p) is equal to

60y '—l-—A=6oy '—/

0A= P26o y '.

(mp + n)4 ^ (7«p3 + np2)4 ^ (m + n'p)4m,ne.£ m.ne£ m,n e.l>

— P2g2(p)- Hence g2(p) = 0, because p2 ^ 0. Similarly, g3(z) equals

= 140 y z t = -140 >7,

(mi+n)6 t—1 (mi2 + ni)6 ^(ni+m')6

mjieZ m,neZ m'.neZ

what equals —g3(z')- This implies g3(0 = 0 (cf [Fo] 67).

There is a well-known result establishing a relation between the weight of a modular

function and its order at the points of H./SL\(Z) (see for example [Sel] VII §3.1 or

[La] 3 §2). From this result, it can be deduced that the only zeros of g2 (resp. g3)are congruent modulo SL2(Ij) to p (resp. i) and are of order 1.


2.2.6 The Discriminant Function A

Let's define A(t) to be the discriminant of the polynomial Ax3 — g2(r)x ~ g?>(*)-

That is

A(r):=g2(T)3-27g3(T)2.

It is a modular form of weight 12, which vanishes only at infinity

A(r) = 7T12 [4096e27"r + ...].

Hence, A is a cusp form and A(oo) = 0 with order 1. q denoting eljtix, A has the

following beautiful product expansion due to lacobi

OO

A(z) = (2jt)12qY\(l~qn)24.n= \

2.2.7 The /-Function

The /-invariant of an elliptic curve E in Weierstrass' form

y = 4x — ax — b

with a3 — 27b2 =£ 0, a, b e C is defined to be

J : =

a3

i3 -21b2'

As seen in §2.2.1, there exists r e H such that, for A := Zt + Z, the map $a :

C/A —> E(C) is an analytic isomorphism. Moreover a = g2(A) = g200 and

b = g3(A) = g3(r). We thus define the /-function on H by

7(t) :=

g2(r)3-27g3(T)2"

It is modular of weight 0 and has only one pole which lies at infinity

J(r) = vfkq-+co+Ciq+-'


where q = e2mix and ct e Q for every i. This implies that 7 takes on each

value once and exactely once in a fundamental domain of SX2OZ) and that J :

H/SL2(Z) -> C is an isomorphism. As seen in §2.2.3, M/SL2ÇZ) is in bisec¬tion with the set of isomorphy classes of elliptic curves over C. Hence two ellipticcurves m Weierstrass' form over C are isomorphic if and only if their /-invariant

are equal.

Remembering that g2(p) = 0 and ^3(2) = 0 with order 1 (cf §2.2.5), we get

/(/>) = 0 and 7(i)=l

with order 3 and 2 respectively. As seen above, 7 (00) = 00 with order 1.

Remark 43. Let H* := HU {00}. 5L2(Z) still acts on M*. The quotient Hl*/SL2(Z)can be equipped with a structure of Riemann surface, for which it is compact. 7

extends then to an isomorphism (renoted 7)

7 : W/SL2(Z)-Z*P1(.C) ~ C U {00}.

2.2.8 The Dedekind ^-Function

The ^-function was defined by Dedekind (1877) to be

00

rj(r) :=q*Y$l-qm),

where q = elnix and for tel. This infinite product is absolutely convergent since

Irnix) > 0 and converges uniformely on every compact subset of EL It therefore

defines an analytic function of r el. All terms of the product being dinstinct from

zero, r] does not take the value zero on H. The product expansion of the discriminant

function (cf §2.2.6) induces the relation

A(r) = (27r)12^(r)24. (2.2)

Since A is modular of weight 12, so is rj(r)24. Therefore, for every x e H and

Y = {acbd)eSL2<&,



and there exists a 24-th root of unity e depending only on y such that, for all reH,

r\(y x) — ev'ex + d r](x).

Once a branch of the squared-root being fixed, the constant e may be determined

explicitely in dépendance of a, b, c, d (see [Ra], Ch 9).

Particular cases are the transformations under the generators of SL2ÇZ) for which

the following holds for every rel

t)(t + 1) = en rj(x),

ï]{-$ = V^ïx Ï](X)

(the branch of the squared-root being here positive for real positive arguments).

Remark 44. The first property could also have been derived directly from the defi¬

nition of 77 (r). More generally, one gets this way, for c e Z and tel,

nie

ï]{x + c) = e 12 77(t).

Remark 45. Given a subgroup r of finite index in SL2ÇZ1), extending the definition

given in §2.2.4, one can define modular forms with repect to r and of real degree—r. Roughly speaking, such functions are meromorphic functions / on H which

satisfy, for each reH and y = (^J) e T,

f{yx) = v{y){cx+d)rf{x),

for a fixed branch of (cx+d)r and a complex number y (7) of module 1 independentof x. Moreover, / shall have at most finitely many poles inXflH, where !R is a

fundamental domain for r, and be meromorphic at each parabolic point of 01. (Notethat this is independent of the choice of Ji.) Such a modular form is called entire

if it is holomorphic in H and at each parabolic point of some fundamental domain.

Moreover, if it has a zero at each parabolic point, it is called a cusp form. (For a

precise definition, see [Kn] Ch2.)

The above shows that 77 (r) is an entire modular form of degree — i with respect to

the full modular group 5X2 (Z). The développement of r\ at the only parabolic point00 of SL2CZ) is

00

rç(T) = eTr J2 (-l)memm(-3m+VT.m=—co

2.3. The Method ofBeukers and Wolfart 111

Setting n := 3m~+m; we see that the exponents are all of the form 2nix{n + -L)with n + jj strictly positive. Hence 77(00) = 0 with order ^ (cf [Kn] Ch 3.3).

2.3 The Method of Beukers and Wolfart

In this section, we explain and generalize the proof of Proposition 2.1 of [BW86] to

a more general solution of the differential equation (2.3) satisfied by the periods on

the universal elliptic curve Ej.

For rel, let's consider the elliptic curve Er over C given by the following equa¬

tion

y2 = 4x3 -g2(r)x-g3(r).

Since the modular /-function is an invariant of the isomorphy class of Ex, let's look

for a représentant Ej of this class which depends only on / = /(t). As seen in

§2.2.7, the /-invariant of the curve y2 = 4x3 — ax — b is / = 3" ,2. Requieringa = b =: c, the equation will only depend on /. For / ^ 0, 1, we have

c3 27/3 =^-^-^c

=

T^x

and Ej is given by

-o , 27/y2 = 4x3-y-T(x + i).

The periods on Ej are functions of /, which satisfy the differential equation

d2Q l dQ 31/-4+ 777+

,AATi,r 77^ = °' (2-3)dJ2 J dJ U4J2(J

-

l)2

as was shown by Klein and Fricke in [KF] II §3. Now, since for each rel such that

/(t) = /, the curve Er is isomorphic to Ej, the lattices AT and A/ correspondingto Ex and Ej respectively (in the sense of §2.2.1) will be homothetic. We intend to

calculate the factor of homothety.

As seen in §2.2.1, a basis of A/ is given by

/dxy

1,2,y

2),

112 2. Identities for some Gauss'


where T>\, D2 form a basis of H\(Ej(C), Z). Let <px : Er -> £> be the iso¬

morphism and Ci, 62 a basis of H\(Er(C), Z), then the push-forwards (<pT)*Cz>i = 1, 2, will form a basis of #1 (£7 (C), Z). Noting that

277 £93'-1 *?'

we check that the isomorphism cpr is given by

,-

-^ ,g2(r) ,82(f).% ,

g3(T) g3(T)

This implies, for ï = 1,2,

dx f dx g3(?) f dx- = / Vr(-)

y JT'

y'

V g2(r) y y

Hence

Aj = //,(t)At,

with

(2.4)

, ,S3(r)

Y S2(T)

The determination of the squared-root does not play any role, because x is fixed and

for any lattice A, we have —A = A. Since 1, x also generate AT,

/x(t) and t/x(t) (2.5)

are periods on Ej. But because of the squared-root and of the multivaluedness of

the association J \-> x, /x(t) and t/x(t) are multivalued functions of J. As such,

they satisfy equation (2.3) and, since they are C-linearly independent, their branches

furnish local systems of solutions.

As we do better understand the /-function than g2 and #3, let's try to get rid of these

ones. To this aim, we work out the following identity for fi(r).

2.3. The Method ofBeukers and Wolfart 113

Identity 7.

m = M

= fe(r)^)i (by (2.4))

= (/(T)ï^=i)4A(T)ïi (by definition of J).

These are multivalued functions of r, whose domains of definition and determina¬

tion will be discussed later.

On the other hand, suppose that we have a solution to (2.3) of the form

Ja{\-jfF{a,b,c;z{J)),

where a, ß & Q and z(J) is one of the following transformation of C preservingthe set {0, 1, oo}

1 1 J J-IJ, -, I- J,

, ,•

J l-JJ-V J

For the moment, we do not wish to bother about domains of definition, indeter¬

minacy and singularity problems and write such a solution in terms of our local

solutions (2.5) considered as multivalued functions of /. There exist A, B e C

such that

Ja{\ - J)ßF(a, b, c\ zU)) = Ar/x(r) + Bp(t).

Using Identity 1 and writing the left-hand member as a function of r, we indeed

loose the multivaluedness due to that of J \-> r, but keep that due to the fractional

exponents. Hence, there exist A, B g C such that

J(r)a(l - J{x)fF(a, b, c; z(/(r))) = (Ar + 5)/(t)"5(/(t) - 1)?A(t)A,

for t to be precised.


2.4 Sixteen Solutions

In this section, we calculate all solutions to Differential Equation (2.3) of the form

G(J) :=Ja(l-J)ßF(a,b,c;z(J)),

with a, ß 6 Q and

1 1 J J-IZ(J) 6 u

7.! - j, —, —, _}. ao

These are called Kummer's solutions. We then recall some theoretical facts on

linear differential equations of the kind of (2.3) together with the ideas of Riemann

to characterize their solutions and of Kummer to find all solutions of this form.

Finally, we study again Equation (2.3) with Riemann's and Kummer's tools.

2.4.1 List of Sixteen Solutions

In order to determine for which a, ß,a,b, c e Q, G (J) is a solution of Differen¬

tial Equation (2.3), we use the fact that F(a, b, c; z) satisfies the hypergeometricdifferential equation

,,c a + b + 1 ab

u"(z) + [— - —- ]u'(z) - — -u(z) = 0.z(l-z) 1-7 z(l-z)

This implies that G(7) satisfies the following differential equation

Q"(J) +Q'(J){-2aJ-1 + 2ß{\ - jy1 - z"(z'rl + jfaz' - ^^z'}+Q(J) {-a(or - I) J-2 - ß(ß - 1)(1 - J)~2 + 2a2 J~2 + 2ß2{\ - J)

-2aßJ-\l - /)-i +z"(z')-1(a/_1 -/S(l - J)~l)

7

^(aJ-1 -ß{\- jy1) + ^^(aJ-1 - ß{\ - JTl)ab

z(l-z)^vfeO2}= o.

For each value of z (J), we require the coefficients of this equation to be respectivelyequal to those of Differential Equation (2.3) and resolve with respect to a, b, c, a, ß.We get the following sixteen solutions.

2.4. Sixteen Solutions 115

7""6(1

75(1

7~5(1

75(1

75(1

7~5(1

75(1

7-5(1

7-i(l-

7"5(1 -

75(1-

7-5(1

7-5(1

7_5(i

75(1-

1 1- jyFi¬

ll

i 5-7)?F(—;

v12

3 7-7)4F(—

12

3 11-7)3F(—

12

i 1

-7)4F(—v12

i 5- 7)?F(—

12

7)4F(

7)?F(

7

n

h

Ï2

i l

-7)4F(—v12

3 7-7)4F(—J

V12

i 1

-7)5F(—7V12

i 57)-5F(—

12

i 1-7)4F(—

12

7)4F(

7)5F(

7)-5F(

7

12

1

12

5

12

1

12

5

12

7

12

11

12

1

12

5

12

7

n

h

Ï2

5

12

11

12

7

12

11

12

5

12

11

12

7

12

11

12

2-.j)3

;7)

4

5; J)

1

2;

1

2'

3

2;

3

2;

i;t)

1;

7)

7)

7)

7)

7'

1

-7

1 7-1

)

2' 7

3 7-1

2' 7

2 7

3' 7-1

4 7

3 7-1

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

116 2. Identities for some Gauss' Hypergeometric Series

Remark 46. The two solutions treated by Beukers and Wolfart [BW86] are (13) and

(15).

2.4.2 Linear Differential Equation of Order Two,

Riemann Scheme and Rummer's 24 Solutions

Linear Differential Equations

Let's first recall some facts on homogenous linear differential equations of order

two ([Ah] Ch8 4)

ao{z)u" + a\{z)u' + ai{z)u = 0, (2.7)

with single-valued analytic functions at for coefficients. Writing (2.7) in the form

u" + p(z)u' + q(z)u = 0, (2.8)

the nature of local solutions will be determined by that of p and q.

Definition 2.5. A point zq is called an ordinary point of (2.8) if p and q are ana¬

lytic in a neighbourhood of zo-

It is known that in the neighbourhood of an ordinary point, there exist two linearly

independent solutions, which are analytic in a neighbourhood of this point ([Ah]

Ch8 §4.1).

Definition 2.6. If p or q has a pole at the point zo, zq is said to be a singular point

of equation (2.8).

There are different kind of singular points. The ones that interest us are the regular

singular ones.

Definition 2.7. The point zq is called regular singular, if p has at most a simple

pole at zq and q at most a double pole at zo-

If zo is a regular singular point, the differential equation has solutions about zo of

the form

u(z) = (z-zoTg(z), (2.9)


where a e Q and g is analytic at zo and satisfy g(zo) # 0. Write

p(z) = —z—+ ...and q(z) = -—^—^ + ...

z - zo (z - zor

for the developments of p and g at zo- Then, if (z — Zo)ag(z) is a solution at zq of

Differential Equation (2.8), the fact that z0 is regular singular implies the followingcondition on a

a (a — 1) — p-\a —

q-2 = 0.

This quadratic equation is called the indicial equation and its solutions a\, a.^ the

caracteristic exponents to (2.8) at zo- The two solutions (z — zo)aigi(z) and {z —

zo)a2gi(z) are linearly independent if and only if «2 — a\ <£ Z. In the case of an

integral difference ai — oc\ > 0, the existence of a solution corresponding to ai

(involving a logarithm) can still be stated ([Ah] Ch8 §4.2).

If the coefficients uq, a\, ci2 are polynomials, we are interessed to know their be¬

haviour "at infinity" ([Ah] Ch8 §4.3). This can be done either by going to ho¬

mogenous coordinates (zo : Z\) with z = f- or by setting z = —. The nature of

the original equation at infinity will be that of the new one at respectively zo = 0

or w = 0. Actually, this amounts to compactify the domain of definition of the

equation, in going from the complex plane to the (complex) projective line.

The first non trivial linear differential equation of order two is that with three reg¬ular singularities ([Ah] Ch8 §4.4). Since any linear transformation of the variable

preserves the type of the equation and the character of its singularities, we can sup¬

pose the singularities to lie at 0,1 and oo. Let a\, a.2, ßi, ßi, Yi, Yi be the pairs of

caracteristic exponents at 0,1 and oo respectively, the fact that the singularities are

regular implies the relation

«i +a2 + ßi +ß2 + Yl + Y2 = 1 (2.10)

and that Differential Equation (2.8) can be written in the form

/,,/! -<*i -«2.

l-0i -ßi\,

u + —— 1 u

V z z-1 )

(a\a2 u\tt2 + ß\ßi - Y\Y2,

ßißi >

+{-i zV^T)

+ Jz^WU= 0-

(2.11)

Let's now assume that

al — a2, ßl — ßi and y\—

yi £ Z. (2.12)


Then, in the neighbourhood of each regular singular point, there exist two linearly

independent solutions of the form (2.9) and Differential Equation (2.11) can be

reduced to

„ ,'c 1 — c + a + b\.

abu" + - H ; I u' + — —u = 0

sz z — \ J z(z — 1)

or equivalentely

z(l - z)u" + [c - (a + b + V)z\u' - abu = 0,

where

a = a\ + ß\ + y\, b = a\ + ß\ + Y2 and c = 1 + a\ - a2.

Since (2.10) implies ß2 — ßi = c — a — b, condition (2.12) translates then to

c — 1, a — b, a + b — c ^ Z. (2.13)

This is the well-known hypergeometric differential equation whose pairs of car-

acteristic exponents at 0,1 and oo are respectively

0, a2-cL\, 0, fo-jSi, yi+ai + j8i, y2 + ai + jßi,

i.e.

0, 1 — c, 0, c — a — b, a, b.

Its solutions are equal to those of (2.11) multiplied by z"ai {z — l)-^1. The hyper¬

geometric series F{a, b, c; z) is the holomorphic solution at 0, where it takes the

value 1.

Riemann Scheme

Riemann was very keen on the idea that a complex function could be characterized

by its behaviour at its singularities ([Ah] Ch8 §4.5). To a linear differential equa¬

tion of order two and with regular singularities at 0,1 and oo, such as Differential

Equation (2.11), he associated the later so-called Riemann scheme

0 1 oo 1

«l ßl Y\

oti ßi Yi

(2.14)


It is unique up to permutation of the columns and of the caracteristic exponents

at a same point. As seen above, such data characterize the local solutions to the

equation. Riemann has denoted the set of all local solutions to Differential Equation

(2.11) by the symbol

P \

With this notation, we can nicely summarize what we have just done in reducingDifferential Equation (2.11) to the hypergeometric differential equation. Indeed,

we have

0 1 oo

a\ ßl Y\

«2 ßl n

0 1 OO 0 1 oo

U\ ßl y\ ; z • =zai(l--z)ßlP 0 0 y\ + a\ + ßi ; z

0L% ßz 72 a.2 — oci h-ß\ y2 + ai + ßi

It can be shown that, to each Riemann scheme (2.14) satisfying condition (2.10),

there exist a unique linear differential equation of order two with the given charac¬

teristic exponents at 0,1 and oo (see also [IKSY] 2.1.1).

Kummer's 24 Solutions

Let's consider the following hypergeometric differential equation

ab0,

( n""' (2-15)

z(z - 1)

and denote it by E(a,b,c). We have already mentioned that F (a, b, c; z) is the

solution at 0 with caracteristic exponent 0. Since 1 — c is the second caracteristic

exponent at 0, there exists a solution to (2.15) of the form zx~~cg(z), with g analyticat 0 and g(0) ^ 0. Inserting it in (2.15), one finds that g satisfies a hypergeometric

equation with caracteristic exponents

c-1,0, 0,c

at 0, 1, oo respectively. Hence

0

a — b, a+1 — c,b + I

G P C-\

0

1

0

c — a — b

oo

a + 1 -

b + 1-

c

c

;z



i.e. g is a solution of E(a + 1 — c, b + 1 — c, 2 — c) and since it is holomorphic at

0 and g(0) ^ 0, we can set

g(z) = F(a + 1 - c, b + 1 - c, 2 - c; z).

In the same way, one can find other solutions to (2.15), only by exchanging the

points 0, 1, oo and requiering that the only two caracteristic exponents equal to zero

correspond to the points 0 and 1 respectively. This means that the hypergeometricdifferential equation (2.15) is transformed into another hypergeometric differential

equation. We present here Kummer's idea to this aim and follow the exposition of

[IKSY] 2.1.3.

First remark that the full group of permutations of the set {0, 1, oo} is

H := AutCPt - {0, 1, oo}) = {mz,-,l-z, ^—, —, -^-}.z 1-zzz-l

To each h e H, we can associate a not unique vector /x = (/xo, Ml, Moo) satisfying

Mo + Ml + Moo = 0 (in order for the sum of all caracteristic exponents to still be

equal to 1, cf Condition (2.10)) such that

/i(0) Ml) Ä(oo)0 + /x/i(o) 0 + /x/z(i) a + M/z(oo) •

,

1 - c + /Xfc(o) c - a-b + Hh(i) b + /x/i(co)

is in the form E{a', b', c') for some a', b', c' 6 C. In our above example, we had

h = identity and /x = (c — 1, 0, 1 — c).

To each h, there correspond actually exactely four vectors /x, since there are four

ways to annihilate two among the four caracteristic exponents corresponding to¬

gether to 0 and 1 (/Xoo being then determined by the condition /xo + Ml + Mco = 0).

This produces 24 solutions to (2.15), each of which involving a hypergeometricseries. These 24 solutions are called Kummer's 24 solutions.

Writing uzo (z, a) for the solution at zq with caracteristic exponent a, we tabulate the

24 solutions. The first four equalities hold because these functions are holomorphicsolutions at zo = 0 having constant term equal to 1 and we know that there is

only one solution to (2.15) with this property. These first four equalities imply the

subsequent ones.


uq(z, 0) = F(a, b, c; z)

= (1 - zY~a~bF{c -b,c-a,c\ z)

= {\-zTaF{c-b,a,c-^-l)= (l-z)-bF(c-a,b,c;Î)

uo(z, 1 - c) = zl~cF{a — c+ l,b — c+1,2 — c; z)

= zl~c(l - zY~a~bF(l -a,l-b,2-c; z)

= z!-c(l - z)c-a-lF{\ -b,a + l-c,2-c;^)= zl~c(l - zf-b-lF{\ -a,b+l-c,2-c;^)

ui(z,0) = F(a, b, a + b — c + 1; 1 — z)

= zl~cF(b + l-c,a+l-c, a + b+l-c;l-z)

= z~aF(a + l-c,a,a + b+l - c; ^)= z"èF(ô + 1 - c, b, b + a + 1 - c; 2=1)

«i(z; c - a - £) = (1 - z)c-fl-èF(c - a, c - b, c + 1 - a - b; 1 - z)

= z1_c(l - z)c~a-bF{\ -a,\-b,c+l-a-b;\-z)

= zb~c(l ~ zY~a~bF(l -b,c-b,c+\-a-b\*=±)

= zfl"c(l - zY~a~bF{\ -a,c-a,c+l-a-b; ^~)

Uoo(z;a) = z~aF(a, a - c + l,a+ 1 — b; \)= {-z)b~c{\ - z)c-a-bF(l -b,c-b,a+\-b;\)= (1 - zTaF{a, c-b,a+l-b;^)= (-z)1_c(l - zY~a~lF(a + l-c,l~b,a+l-b;-±-z)

Uoo(z;b) =z-bF{b,b-c+\,b+\-a;\)= {-z)a-c{\ - zY~a-bF{\ -a,c-a,b+\-a\\)= (1 - z)~bF(b, c-a,b+l-a;j^)= (-z)^c(l - zY~b~lF(b + l-c,\-a,b+l-a;j^-z)


2.4.3 A New Eye on our Solutions

Since Differential Equation (2.3) has only three singularities, which are regular and

lie at 0,1 and oo, we can find some solutions by Kummer's method and with Rie-

mann's notation as described in the previous section.

In order to determine the caracteristic exponents of Differential Equation (2.3), we

compare its coefficients with those of Differential Equation (2.11) and resolve with

respect to ot\, ai, ß\, ßi, y\, yi. We get

1 1

6' 6

3 1

4' 4

0, 0

Hence the Riemann scheme associated to (2.3) is

atO,

at 1,

at oo.

0 1 oo

0

0

Wanting to reduce to a hypergeometric equation, we use

P \

where

0 1 oo

0

0

J • =75(1 -J)iP 0

oo

011

12

1

3

1

2

11

12

/

0 1 oo

0 011

12

1 1 11

3 2 12

12 12 3

Remark 47.

One immediately notes that, since a — b, our equation does not belong to

the general case, where we could ensure the existence at each regular singular


point of two linearly independent solutions of the form zag(z), with g ana¬

lytic and not zero at the point. However, we can still use Riemann's notation

and Kummer's method to get new solutions to the equation. We just have to

pay attention to the fact that this method will not produce two linearly inde¬

pendent solutions at infinity, because the characteristic exponents at infinitydiffer from an integer. This is also the reason why we only have 16 solutions,since the solutions which are symmetric in a and b fall together.

• The orders of the generators of the monodromy group of the hypergeometricequation being the absolute values of the inverses of the characteric exponents

differences at each singular point respectively, we can read them easily in the

Riemann scheme, they read

3, 2, oo.

Let's now tabulate Kummer's solutions to Differential Equation (2.3) (on the right)

by multiplying those to E(^,n11

]i,f) (on the left) by 75(1 J)î:

uQ{J, 0)

P(ll 11 4.js

v12 ' 12 ' 3''

(1 J ) 2-*vÏ2'Ï2'3'''

_

V1 J ) ?vi2' 12' 3' J-\>

/6(1 J)A r (12j 12' 3> •*)

5 4.

75(1-7)?/^,^,*; 7)

u0(J, 1 - c) =

y-ÎF(^, ^, f; 7) 7-5(1 - 7)*F(^, ±, §; 7)

7-5(1 - 7)-2F(^, i, §; 7) 7-6(1 - 7)?F(i i §; 7)

7-3(1 - 7)-ÄF(^, ^ |; 7éT) 7-5(1 - 7)5F(£, i |; ^)

ki(7,0) =

F({±, |i, §; 1 - 7) 75(1 - 7)f F(ii, %,\\l- J)

J~ha, 1 1; 1 - 7) 7-5(1 - J)*F(1 ±, |; 1 - 7)

7~i2F(

12' 12' 2

1 11 2. 7-1

12' 12' 3' / ) /-|(1 _y)|F(7 n 3;Zfi)



u\(J', c — a — b) =

(l-/)-ïF(i,i,I;l-7)

;-5(i-;)-lF(iy;i-;)/-Â(l-/)4F(i è,i;Î)

75(1-7)^(^,1; 1-7)

/4(l-/)^F(i,i,i;l-J)

aoo(7, a) = Uoo(J,b)

U rvll I.12' 12y-T2F( ,i;i)

rn(l-JnF(^,l;i)

/-!(l-/)lF(li,^;l;i)/4(l-/)lF(i,^,l;I)

ll(1 - 7)-nF(^, %, 1; ^ 75(1 - -/r^ïb ïi ^ 1=7)

7-?(l - /)-ÄF(i, ^, 1; y^) 7-5(1 - 7)6F(i, X, 1; ^j).

2.5 Twenty-two Identities

As announced in the introduction of this chapter (Section 2.1), we shall now make

use of our extension (cf Section 2.3) of Beukers' and Wolfart's method in order to

produce at least one identity for each of our solutions to Differential Equation (2.3).

According to Section 2.3, for each solution 7a(l — J)^F{a, b, c; z(7)), there exist

A, B e C such that

7(T)a(l - J(r))ßF(a, b, c; z(7(r))) = (At + 5)7(t)"5(7(t) l)3A(r)ik(2.16)

We have now to choose a domain of définition on which both members of the equal¬ity may be well-defined as holomorphic functions of r. Since the hypergeometricseries converges in the unit disk, the first step is to restrict tor e I such that

|z(7(t))| < 1, that is to work locally at a point to M such that z(7(to)) = 0.

Notation 1. For Xq e {i, p, —p, oo} and z(7) a transformation of 7 among those

of (2.6) such that z(7(to)) = 0, Cz(/)iTo will denote the neighbourhood of To in

{t £i; |z(7(t))| < 1}.

Note that C7-z(j),to 1S connected, because 7 resp. j, jhj behave in such neighbour¬hoods of i, p, —p resp. oo like n-fold coverings (n = 2 at i, 3 at p and —p resp. 1

2.5. Twenty-two Identities 125

at oo). In particular, Cz(/)iTo is invariant under each modular transformation fixing

to, because it is connected and because / is invariant under such a transformation.

If such a modular transformation is applied to Identity (2.16), the invariance of the

left-hand side will imply a relation between the constants A and B. This is the

method we will use to have stronger identities. The solutions in a neighbourhood of

/ = 0, 1, oo respectively will be treated separately.

2.5.1 About J = 0

The solutions about / = 0 are of the form Ja(\ — J)^F{a, b, c; z(J)), with

/ 1 1

z(J) g {/, -—-} and «{---}.7 — 1 66

If t is such that \J(x)\ < 1 or \/(^\\ < 1, then J(x) ^ 1. This implies that,

if z(J) e {J, jzrf} and xq e {p, —p}, then each branch of (J(x) — l)+~^ is well-

defined on Cz(7)jTo as holomorphic function of r. Because A does not vanish on

EI, the same holds for A(r)i2. There is only one term to worry about, which is

J(x)~6~a, but it will appear to be easy to handle, because the only values taken a1 1

are — ^ and ^.

In the Neighbourhood of p

The modular transformation x h> — ^W fixes p and therefore preserves Cz(j)iP.Since J is invariant under such a transformation, so is the left-hand side of (2.16)

and so must be the right-hand side. Hence, the following condition must hold for

all t e Q(/),p and for one 12-th root f of unity.

AT + 5 = f(-4r + 5)(r + l)

O Ax + B = BÇx + Ç(B -A)

O A = ÇBmdB = £(A - B)

=> C_1 = 1 — ç-

^ Çe{-p,-p}

and B = —pA or B = —pA.


Considering our solutions about J = 0, we distinguish two cases.

• Caseha, = \, F(a, b, c\ z(/(t))) = (At+5)/(t)_5(7(t)-1)?-Â(t)Ä.

The left-hand side and (7 — l)?-*0 A12 are holomorphic at p, but 7-3 has a

pole of order 1 at this point. Hence, Ax + B must have a zero of order 1 at p,

that is

B = ~pA

musthold. Then (At+5)7(t)-3 is holomorphic and well-defined on Cz(/)iP.

• Case 2:a = -\, F(a, b, c; z(7(t))) = (At + B)(J(r) - 1)Â(t)h.

Since the left-hand side and (7 — 1)3-^ A12 are holomorphic and nonzero at

p, so must At + B be. This implies the relation

B = -pA.

This shows that in both cases, the identity (2.16) is well-defined on Cz(j)tP. Since

this domain is connected, A12 can be replaced on it by rf up to a unique complex

constant. Hence, we have shown that

3A e C such that Vt g C/ij0 we have

F(l2' ~k' \' J(T)) = A(T ~~ ß) r}{t)2 (1)

F(é' ïï' ï'7(T)) = A(T ~p) -w^)2 (2)

7 H 0

F(~, —, -; 7(t)) = A(t - p)(/(r) - l)-^(r)2 (3)

^(~, ^, \\ Jir)) = A(t - p) 7(t)"î(7(t) - 1)-^(t)2 (4)

and 3A C such that Vt e Cj_ •

1 7 2 7(t) 1 9

F(7v 7vV r, \ i> = A<T - ^/(t) - i)11^7) (15)

12 12 3 7(t) — 1

5 11 4 7(t) 1 s,

12 12 3 7(t) — 1


Remark 48. Note that these identities correspond to Kummer's identities listed at

the end of §2.4.2. For example

(2)

(4)

(16)

C/(r)-l)-2 F(^,^,l; J(t))

^12' 12' 3'

(/(r) - I)"ü12

FfX 11 ft. -/(r) n

r ^12' 12' 3' 7(t)-1^ J

= A(T-p)7(T)-

(/(t)-!)-^^

In the case a = — ^, the constant A is very easy to calculate. Indeed, setting x = p

in each identity, we get

Vt e C/,p:

112 t — p n(r) -,

12' 12' 3' iV3 Vp)

7 7 2 t — p i n(r) n

F(-, -, -; 7(r)) = —f(1 - 7(r))-2(^)212 12 3 ;V3 ?7(p)

(2.17)

(2.18)

and Vt g C^_ •

/-i"°

F(1 7 2

12 12 3 7(t)- 1

7 Or) x — p i /?(t) o

) = —7#d-/(T))T2(^)2;V3 ??(p)'

(2.19)

The branches of the roots are here determined by (1)2 = 1 resp. (1) 12 = 1.

For the identities where (r — p)7(r) 3 appears, the constant can be calculated in

dépendance of the residue c-\{J~ï, p) of 7~3 at p. Since J~ï has a pole of order

1 at p, it has a Laurent-expansion at p of the form

C-l(/ 3,p)

T -pÄ(T)

where h(x) = J] c^(/ 3, p)(r — p)* is the entire part and

_i 1c_i(7 5,p) =

27TZ7(z)-ïJz

Z-Pl=r


the residue. The radius r satisfies 0 < r < Rp, where Rp is the radius of conver¬

gence of h(x). Hence, for t e D(p, Rp), the open disc centered at p with radius

_i

Rp, the product (r — p)J(x) 3 equals

C_i(7~?,yO) + (T-yO)/l(T)

and takes the value cî(J~ï ! p) at p. This allows us to calculate the constant A in

each of the identities (2), (4) and (16). We get

Vt e Cy, p-

5 5 4 i , i nix) r.

F(~, —, -; 7(r)) = c_!(/-3, p)-1^ - p)7(T)-7(-^)2 (2)12 12 3 r?(p)

11 11 4 1 , 1 1 17(t) o

F —, 777- ^: -/(t)) = c_!(/", /o)_1(r - p)/(r)-3(l - /(t))^^)212 12 3 ??(p)

(4)

andVr e Cj_j-i'P

5 11 4 7(t) 1 t1 5 j7(t) 9

12 12 3 /(t) — 1 77 (p)

(16)

The roots are determined respectively by (1)2 = (l)12 = 1.

In the Neighbourhood of —p

Let's now work in the neighbourhood Cz(j),-p of —p. Since F(a, b, c; z(J(j)))

and (7(t) — 1)4~Â(t)T2 are well-defined and holomorphic functions of r in

Cz(j),~p, so must be (Ax + B)J(x)~-h*.

The case a = ^ is the easiest one. Indeed, in this case, J~6~a = J~? has a

pole of order 1 at —p. Hence, Ax + B must have a zero of order 1 at x = —p,

i.e.

B = pA


must hold. Under this condition, (At + B)J(x) 3 is well-defined and holo-

morphic on Cz(j). -p-

• In the case a = —

g,J~~s~a does not appear in the identity and no holomor-

phy condition implies any relation on A and B. Remarking that the modular

transformation r h>- 1~- fixes —p and preserves Cz(j)-p and applying it to

both members of the identity

F(a, b, c; z(7(t))) = (At + 5)(7(t) - 1)?"Â(t)A, (2.20)

we get the following condition for the right-hand side to be invariant. It must

be valid for all r in Cz(j)-p and one 12-th root Ç of unity.

At + B = (A^ + B)Çt

4> Ar + B = Ç(A + B)r-ÇA

<£> A = ÇA + ÇBmdB = -ÇA

^ l + f2 = £ and£ = -fA

<$ Ç 6 {-p, -p}mdB = -ÇA

<$> B = p or B = pA.

Since the left-hand side of (2.20) is nonzero at —p, so must be the right-handside, hence

B = pA.

Our identies are valid on Cz(j)-p. On such a connected domain, AT2 can be re¬

placed by rj1 up to a unique complex constant. Then

3A e C such that Vt e Cj--p:

F(~, ^, ^; 7(r)) = A(t + pMrf (V)

F(-, —, -; 7(t)) = A(r + p)7(t)"^(t)2 (2')

F(—, —, -; 7(t)) = A(r + />)(7(r) - 1)~^(t)2 (3')

F(li li ^; J(T)) = A(r + ^/(T)"^/W - irW)2 C4')


and 3A e C such that Vr e Cj_ _-:

j-i' p

1 7 2^(—>t^~;

F(

12 12 3 7(t)-1

5 11 4 7(t)

/(T)) = A(t + p)(7(t)-1)^(t)2

.'

* /x'

/*'

12 12 3 7(r)- 1) = A(r + p)/(T)-3(/(r)-l)T5^(rr

(15')

(16')

In the cases where a = —

g,the constant A is easy to calculate. One gets

Vt e Cj,--P:

F1 1 2 T + p 7?(T) 2

7l—, —. -; 7(t)) = —-( Y

V12' 12'3' iV3 »î(-p)

7 7 2 T + p i ?7(t) ?

F(-, -, -; 7(t)) = —f(1 - 7(t))-ï(-^t)

12 12 3 zV3 »7(-p)

and Vt e C j - :

1 7 2 7(T)

12' 12'3' 7(r)-r /V3

T + p 1 n(r) 9

)-^1(l-7(T))T2(-^4-)2

?7(-p)'

(2.21)

(2.22)

(2.23)

Again, the the branches of the roots are here determined by (1)2 = 1 resp. (1) 12 = 1.

Note that rj(-p)2 equals -ipr)(p)2 (cf §2.2.8).

For the identities where (t + p)7(t)~3 appears, the constant can be calculated in

dépendance of the residue of 7" at —p. Let R-p > 0 be such that, for t in

D(—p, R-p), we have

T + P

and the entire part g(x) = J2k>o ck(J~^> —p~)(j + ß)k converges in D(—p, R-p).The residue at —p is

_i_

1

c_i(7 3,-p)2jri

\z+p\=r

dz

7(z)'


whereO < r < R^ß. The product (x+p)J(x) 3 takes then the value c-\{J 3,—p)at -p. Setting x = —p in Identitites (2'),(4') and (16'), we get

Vr e Cjt--P:

5 5 4 i i i ?7 (t) nF(—, —, -;/(r))= c-i(J-i,-p)-\t + p)/(r)-3(-il4-)2 (2')

12 12 3 r)(—p)

11 11 4 1! 1 1 7}(T) o

F(—,—>-;7(T))=C_1(/-ï)-p)-1(T + yô)/(T)-Ml-/(T))-5(-7L^)12 12 3 rj(-p)

(4')

and Vr g C /

7-1 .-P'

5 11 4 /(t) it is r?(r) ,

F(-!-,-;77^T)=c_1(/--ï)-pr1(T + p)/(rr3(l-/(t))T2(^4-)212 12 3 7(r) — 1 ?](—p)

(16')

The branches of the roots are determined respectively byl2 = ln = l.

2.5.2 About / = 1

We here deal with solutions Ja(l — J)^F(a, b, c; z(J)) where,

7-1 13z(J){l-J,——} and ß e {-,-}.

We here choose to work in the neighbourhood Cz^j of x = i. Since / and

A do not take the value 0 on Q(/)/, a unique determination of J(x)~^~a resp.

of A(t)ï2 can be well-defined as holomorphic function of x on C^/y. Since

F(a, b, c; z(J(x))) is holomorphic on Cz(j)j, (Ax + B)(J (x) — l)"*0 must also

be so.

• In the case ß = |, (J(x) — l)*-*8 = (J(x) — l)-2 has a pole of order 1 at i.

Hence, Ax + B must have a zero of order 1 at z, i.e.

B = -iA

elation, (

holomorphic on Cz(j)j.

must hold. With this relation, (At + B)(J(x) - 1) 2 is well-defined and



In the case ß = |, (2.16) reads

F(a, b, c; z(7(r))) = (At + B) J {xyÂ{r)^-

and both members are holomorphic on Cz(j)j without condition on A and B.

But a condition can still be obtained by applying the modular transformation

t i-> —ion Cz(j)tl. This transformation indeed preserves Cz(j)yl, because it

is continous and fixes i, while Cz(j)j is connected. The invariance of / under

this transformation implies that of (At + B)A(r)n. Hence, the following

condition must hold for all x G Cz(y)i( and one 12-th root £ of unity

At + B = £(-£ + B)x

& Ax + B = -ÇA + ÇBx

4> A = £5 and£ =-£A

=» r1 = -£

^ f e {±z}

and 5 = z A or 5 = —iA.

i

Because J and F(a, b, c; z(J)) are nonzero at i, so must be (At + 5)A(t)i2

and the following relation must hold

B = iA.

In each case, we have shown that the identity is valid on Cz(j)j. Since this domain

is connected, À12 can be replaced on it by r\ up to a unique complex constant.

Hence, we have

3A g C such that Vt e Ci_/.,-:

F(Î2' h 2'l ~ J(T)) = A(T + i)ri{X)2 (5)

F(T2' n'b1' /(T)) = A(T + z')7(rrW)2 (6)

F(T2' 12' 2;* " /(T)) = A(T ~ °(/(T) " ir^(T)2 (7)

11 11 ^

F(—, ^^l~ J^)) = Mr - i)(J(r) - 1)-5/(t)-^(t)2 (8)


and 3A e C such that Vt e Cj-\ .:

j >l

F(1 5 1 /(t)- 1

12' 12' 2' /(t)) = A(t + /)J(t)T2^(t)^

7 11 3 7(t)-1i 7 ,

F(—, —, -;-^ )

= A(t -

i)(/(r)- 1)_2 J(T)T2^(r)2.

v12 12 2 /(r)' ^ n w y v /v

(13)

(14)

When ß = 1, it is easy to determinate the constant A. Indeed, setting x = i in

Identities (5),(6),(13), we get in each case A

Vr e Ci-jr.

lu-] ayand, eventually,

111 x + i n(x) n

V12 12 2V !!

2i r]{i)

5 5 1 x + i \ n(x) o

F(—, -, -; 1 - 7(r)) = —^/(t)-ï(-^)212 12 2 2z 77 (z)

and Vt e C/^ :

(2.24)

(2.25)

1 5 1 /(t)-1F(— — -

—— )12' 12' 2' 7(t)

x + i i ri(x) o

—/(r)^(^_Z)22z '»7(0

(2.26)

The roots are here determined respectively by (1) 3 = 1 and (1) 12 = 1.

Remark 49. Identities (13) and (15) are stated respectively in Proposition 2.1 and

in the last remark of [BW86].

In the other cases, the constant can be determined in dépendance of the residue of

(/(t) — 1)~2 at t = i. Since J(i) = 1 with order 2, (/(r) - l)" has a pole of

first order at i. Then there exists Rt > 0 such that, for x e D(i, R{),

(7(r) - 1)" =—^ '—^

+ f(x),x — 1

where the entire part f(x) is convergent in D{i, Rt) and

c-MJ -iyU = Ù I\z—i\=r

dz

(7(z) - 1):


for any r satisfying 0 < r < Rt. The product (t — i)(J(x) — 1) i takes then the

value c-i((J — 1)~2; j) at i. This implies, in each case among (7), (8), (14), that

A = c_i((7 - l)-ï, z')~V0~2- Hence

VtCi_7iJ:

7 7 3 i 1 i n(t) ,

F(-, -, -; W(t)) = c-iC^-l)^, 0_1(r - i)(J(r)-\rH^-)2 (7)12 12 2 r](i)

11 11 3 i i ii ?i(r) 9

F(—, —, -; 1-7(r)) = c_!((/ -l)-5, i)-1^ - i)(/(T)-l)~3i(t)-3(^T()212 12 2 r](l)

(8)

and Vt e Cjî,

:

7 11 3 7(t)-1 i iI? n(T) ,

F(-, -, -; 4tM = c-i((/-ir*, 0_1(r - i)(J(r)-l)"i(r)T2(^)2.12 12 2 7(t) j?0)

(14)

2.5.3 About J — oo

We consider here the solutions Ja(\ — J)&F(a, b, c; z(7)) with

z(/)G{7'T37}anda + ^ = 0-

The functions 7 and A are invariant under the modular transformation x t-> t + 1

and admit a Fourier expansion in q := g27"r (cf §2.2.4) which is called expansion

at infinity. These are given in §2.2.7 and §2.2.6 respectively and show that A (resp.7 and 1 — 7) has a zero (resp. pole) of order 1 at infinity. The application of the

modular transformation r m- r + 1 to Identity (2.16) and the invariance of 7 and A

under it show that Ax + B must also be invariant, hence that

A = 0.

One verifies that B J {x)~~î~a {J {x) — l)4~Â(T)n has then a pole of order—-g

—

a + | — ß —

j2= 0 at infinity, because a + ß = 0. This corresponds to the

holomorphy of F (a, b, c; z(7(t))) at oo. Hence, both members are well-defined


and holomorphic in Cz(j)<00. Replacing A12 by r)2 up to a complex constant, which

is unique in Q(/)j00, we have

3B e C such that Vr e Cx

15 1 19

F(—, —, 1; ) = BJ(x)v-n(r)212 12 /(r)

7 11 1

v12 12 /(ry

(9)

(10)

and B5 e C such that VreC1-7'

^(ttt, 7^> i; -,—t^t) = WOO - i)^00212 12 1-/00'

^(5 11

l;1

12 12 1-/(t)) = 5/(r)-3(J(r)-l)T2,?(ry

(ID

(12)

Since a + ß = 0 in each case, the constant term of the development at infinity of

/-§-«(/_ l)4"Vis(1728- \-a+\-ß,-\ _

case, B = 172812 = -^12. Inserting this value, we get

) = 1728 12. This implies that, in each

Vt g Ci

15 14^- 1

F(—, —, 1; ) = V12/(t)î2?7(t)V12 12 j(xy

v ; /v y

7 11 1F(— — 1- )M2' 12'

'

J(rYyÏ2J(x)u(J(x)-l)~ri(x)2

(2.27)

(2.28)

and Vt e C1-/'

1 7 1F( 1- ) =M2' 12'

'

l-/(r);^12(/(r)-l)n?5(r)2

5 11 1Ff 1- ) =

12' 12''

1 - /(r)-yÏ2/(T)-ï(/(r)-l)A^(T)2

(2.29)

(2.30)


2.6 Some Applications of our Identities

2.6.1 Algebraicity of some Hypergeometric Values

Applying some classical facts about elliptic functions to some of our identities,

we will deduce the algebraicity of the values of the corresponding hypergeometricseries at certain points.

Let us first recall two facts:

1. Let A, B, D e Z with AD > 0 and for x e H, set <p(x) = (^"TT^2, then

the function <p is integral over Z[1728/] (see [La] 12§2 Thm 2).

2. If t e His imaginary quadratic, then J(x) is an algebraic integer (see [La]

5§2Thm4).

Proposition 2.1. Let x e Q(z') n C\-jj, then 1 - J{x) e Q,

and1 1 1

F(—,—,-; 1 -/(t))12 12 2 ^4,i;l-7(r))e

Proof Let t e Ci-jj nQ(j'). The first asseition follows from the second fact, since

Im(x) > 0 and Q(z') is a quadratic extention of Q. Further, there exist A, B, D e Z

such that t = ii^-. Note that Im(x) > 0 implies AD > 0. By the first fact,

( £ )2 is then integral over Z[1728/(z)] = Z[1728] and, in particular, algebraic.Now, it remains to apply this to Identities (2.24) and (2.25) respectively. D

Proposition 2.2. Let x e Q(z') n C/=i -,?/zen J^hl e Q <277 0,

the algebraicity of ^=^- and (,JQ)2 by the two facts. If x e Q(z') n Cj-i .,we

J (T) ]][}) j ,1

conclude by using Identity (2.26). D

2.6. Some Applications of our Identities 137

Proposition 2.3. Let x e Q(iV3) n (CJiP U Cj--p). Then J(x) g Q,

and1 1 2

F<12' 12'31/W) e

7 7 2F(—,—,-; Jit)) G

12 12 3

Proo/ By the second fact, / (t) e Q follows from [Q(i V3) : Q] = 2 and Im{x) > 0.

If t e Q(i a/3) n CJiP, then 3A, 5, D G Z such that t = ^^ (because Q0'a/3) =

Q(/o)) and /m(r) = 4^ > 0. Thus AD > 0, so that we can apply the first fact

'2i(T)\2to(r^4)"

with Z[1728/(p)] = Z. In this case, we conclude by using Identities

(2.17) and (2.18) respectively. In the case where x G Q(z"a/3) D Cj-s, using that

:i>/3) = -p), we can write x as

Jl(l)_)2D

with A, 5, D g Z and AD > 0,

hence ( , -.

with the fact that J{x) G

Proposition 2.4. Le£ r G

G Q. We conclude by using Identitiy (2.21) resp. (2.22) together

D

(iV3) n (C^_„

u c_^ _-), then/(r)_1

anJ

1 7 2 7(T)

12 12 3 /(r)- 1)e

Proof. By the second fact, for r g Q(z'a/3) with Im(x) > 0, we have jz^ßi e Q.

Moreover, we have shown in the proof of Proposition 2.3, that, for such a x,(^f4)2

and (J_L)2are algebraic. If t g

rç(p)'

^)j(-p)'

(2.19), while if x g

z a/3) n Cj_ ,the result follows from Identity

y-i

(i V3) n C^t_^_ß, it follows from Identity (2.23). D

Remark 50. In [BW86], the result of Proposition 2.2 is asserted for each x eQ(i)nH such that \^j^y-\ < 1 (cf Thml) and that of Proposition 2.4 for each x in

i V3) n H (cf Thm2) such that7(T)-1

< 1.

2.6.2 Some Algebraic Evaluations

Using some long known results on modular equations (see [We]), we will calculate

the values of J (mi) and 11 (mi)

»7(0for some m G N. These values will be found to

be applicable only in the identities where z(J) = ^f-, because this will be the


only transformation z(J) among {/, 1 — 7, y^y, j^} for which \z(J(mi))\ < 1.

Inserting these values into Identity (2.26), we will get some values of F(^, j^, \; z)which will be algebraic, showing agreement with Proposition 2.2.

The nontrivial calculations of this paragraph have been made using Mathematica.

Gathering of Datas

The functions ofreH

f{x) = e 2*—?— and /1(r) = —\77 (t) ï)(x)

satisfy the following relations with the /-function

(/(t)24 - 16)3JM =

2«33/(r)»' (231)

,(t) = </-'f4+'f,p.32)

2633/!(r)24v ;

t_

(256 + /!(t)24)3/(2} ~ 2633/l(r)48

(2'33)

(cf [We] §34 (9) for the definitions and for the relations §54 (5) and §72 (3), (4)

respectively).

The following datas of Tabelle VI, pp721-726, [We] will also be used.

M2i) = Vs (2.34)

f(3i)3 = ^2(1 + V3) (2.35)

/i(4/)4 = 2^/8(1+72) (2.36)

/(5i) = 8-ï(l + V5) (2.37)

/i (603 = ffix, where x2 - 4x - 2 = 2-73 (x + 1) (2.38)

f{li)2 = Vlx, where x + - = 2 + VÏ (2.39)x

/i(10i) = 2"5X, where x2 - x - 1 = V5(x + 1). (2.40)


It can be shown (cf [We] §72) that the function 27(^t406 satisfies the equation//IT)

x4 + 18x2 + 122(7(t) - 1)2X -21 = 0.

Moreover, 5(,J^j-)2 satisfies

x6 + 10x3 - 12/(t)ïx + 5 = 0,

(2.41)

(2.42)

while 7(^)2 satisfies

x8 + 14x6 + 63x4 + 70x2 + 125(7(t) - l)5x -7 = 0. (2.43)

Calculation of J (mi) and (^f)2 for some m e N

The values of J (mi) for m e {2, 3, 4, 5} is obtained easily by inserting the values

(2.34) to (2.37) into formulas (2.31) or (2.32) respectively. This yields

7 (20

7(30

(83 + 16)3_

(ll)3263383 23

((l + V3)8-4)3_

133283_

237219-31-v/3

2233(1 + V3)8~~ +

32

7(40

7(50

(2T(l+V2)6 + l)3 225-181-210319+ 3472(11)219-59V2

2h3(l + V2)6

((1 + V5)24 - 222)3

25

24233(1 + V5)24= 1637 • 2659 • 2927 + 243472 • 23 • 47 8375.

The value of ^ is given by h (20 = M, hence

(^0)2 = 2_srj(i)

(2.44)

Remark 51. Using the definition of the ^-function (cf §2.2.8), one can verify that,for m,n e N, the quotient j?^- is real and positive.



In view of the above remark, (2.37) implies (^jb2 = 2 s (1 + a/2) 2 and using

(w>2 =®202 tosether with <2-44)' we have

J7(0 2T(1+V2)3(2.45)

According to (2.41), the function 'iij^—)1 satisfies the equation

z12 + m6 27 = 0,

because J(i) = 1. Solving this equation with respect to z, one gets only one real

and positive solution, which is (3 (—3 + 2a/3))s. This implies

rim

77(1))2 = 3"i(_3 + 2^3)5 (2.46)

Similarly, Equation (2.42) has only \(— 1 + V5) as real positive solution. Hence,

(2.47)(r](5i) 2-1+V5

rç(0 2-5

The next step is to find the value of f\ (6z')3. According to Remark 51, we know that

it is real and positive. Since the equation x2 — Ax — 2 — 2s/3(x + 1) of (2.38) has

a positive and a negative real solution, we only need to choose the positive one and

the positive real 8-th root of 8 to get

/i(6i)3 =2t(2 + V3 + V9 + 6V3).

From this, the value of J{6i) is obtained by using Formula (2.32). Indeed,

7(60 =((2 + y/3 + V9 + 6V3)s+2)3

33(2 + V3 + V9 + 6v/3)8

Since M6if =(£§h3,

we get Ä)2 = 2~*(2 + V3 + V9 + 6V^)"f and

together with (2.46)^(6i)' ^(3i)

T](l)

(-3 + 2^3)5

2335(2 + ^3 + ^9 + 673)?

2.6. Some Applications ofour Identities 141

In view of (2.43) and of Remark 51, 7(^ry)2 is a real positive solution of

x8 + 14x6 + 63x4 + 70x2 -7 = 0.

Again, there is only one real and positive solution. It reads

I + -JÏ + W-l + 4V7 and impliles

Now, we would like to know the value of Jili). In order to do this, let's use (2.39)and solve x + - = 2 + V7 with respect to x. The solutions are

-(2 + V7-V7 + 4V7) and -(2 + V7 + yjl + 4^7)

\2 1and are both real and positive. To determinate which of them equals f(li) 4=, first

consider that

oo

f(7i) = e%Y\(l+e-7jt(2k+V).

In

k=0

-35jta2 1It is sufficient to evaluates i2 (1+ <? /7r)-(l + g 2L7I)-(l+e 337T)/^ to find a veryV2

good approximation of

^(2 + V7 + V7 + 4V7)

This implies that

f(7i)2 = 2-5(2 + VÏ + V7 + 4V7)

and together with Formula (2.31)

J Hi) =((2 + V7 + V7 + 4V7)12 - 210)3

21833 (2 + 4Ï + V7 + 4V7)12

According to Formula (2.33), f\ (Si) is solution of

(256 + z24)3 - 2633/(4/)z48 = 0.

142 2. Identities for some Gauss' Hypergeometric Series

This equation has only two real positive solutions, which are

(2(-140 + 99V2))À and

25? (V • 257 • 2441 + 2232 • 11 • 17923^2

+32y 22 • 3 • 221047 • 937823 + 11 • 17 • 139 67672987V2j"

.

Well, let's try to get an approximation of the value of

CO

/1(8o=^n(i-e_&r(2*+i))-

The evaluation of

ef(l_e-8^)(l-e-2^)(l_e-40ff)

gives a good approximation of the second solution found above. Hence,

{^——)2 = 2"T2 (V • 257 - 2441 + 2232 • 11 • 17923^2rç(4i) V

+32V 22 3 • 221047 937823 +11-17-139- 67672987V2'

and together with (2.45)

(!!Lll)2 = 2~2?(1 + V2)_2 (V . 257 -2441 + 2232 11 • 17923V2r](i) \

+32V22 • 3 • 221047 • 937823 + 11 • 17 • 139 67672987M

Inserting the value of f\ (80 in Formula (2.32), we get

,where

(27T+1)3J (Si) =

2533T


t] := 24 • 257 2441 + 2232 11 17923V2

+ 32a/22 • 3 • 221047 • 937823 + 11 17 • 139 67672987^2

In order to calculate /i(10Q, we refer to (2.40) and solve equation x2 — x — 1 =

-s/5(* + 1) with respect to x. Only one solution is real and positive. It reads j(l +

V5 + V10 + 6V5). Hence, ft(lOi) = 2-s(1 + V5 + V 10 + 6V5) and

2Î

V(5i) (1 + V5 + V10 + 6V5)2

Together with (2.47), we deduce

0 =

rj(i)

25(-l + V5)

5(1 +V5 + V10 + 6V5)2

and with Formula (2.32)

7(10i)((1 + V5 + y 10 + 6V5)24 + 231y

26033 (1 + V5 + /ÏO+6V5)24

Some Algebraic Evaluations of F(-^, ^, \; z)

Considering the results of the previous subparagraph, one sees that z(J) = ^-r-

is the only transformation among {/, 1 — /, ^-, -f^} for which \z(J(mi))\ < 1

when m = 2, 3, 4, 5, 6, 7, 8 or 10.



The following simplifications can be calculated

J{2i) - 1_

1323_

3372

J{2i)~

Ï33l_ Tî3"

J(3i) - 1_

2372(22 11093 - 3 • 19 31>/3)

J(3i) (11 • 23)3

J {Ai)- 1_

3372ll2(83967-24-3- 19 • 59V2)J {Ai) (23 • 47)3

J {Si) - 1 243372(23 • 5 5237 • 22067 - 3 23 • 47 • 83^5)J (50 (11 59 • 71)3

Applying the above results to Identity (2.26), we find the following evaluations of

F{^, j2, \\ J(j(~}1) at t = mi form = 2, 3, 4, 5, 6, 7, 8, 10 respectively.

For t = 2i, we get

1 5 1 3372 3-114F{—,—,^^tt)

12' 12'2' ll3 ll

then, successively, for r = 3i:

1 5 1_ 2372(22-11093-3-19-31V3)12' 12' 2' (11 -23)3

~(3-7 + 22- 5^3)33

r = 4i :

J_ 5 1 3372112(83967 - 24 - 3 19 - 59^2)(Ï2' 12' 2' (23 - 47)3

) ^(7-13+22-3-5v/2)5

t = Si:

1 5 1 243372(23 • 5 • 5237 - 22067 - 3 - 23 • 47 • 83^5)(Ï2' 12' 2' (11-59-71)3

)

-(7-23 + 23-3-5V5)?


t = 6z :

J_ _5_ 1 (2 + (2 + V3 +79 + 6V3)8)3- 27(2 + 73 +V9 + 6V3)812'12'2'

(2 + (2 + V3 + V/9TôV3)8)3

7

22-33 • 7 11 + 22 • 5 7^ + 23 • 3 5J2 • 3 +

7V3

r =7i:

1 5 1_

((2 + V7 +V7 +4V7)12- 210)3- 21833(2 + V7 +y/l + 4V7)1212' 12'2'

((2 + V7 + /7+4V7)12-210)3

— (7-43 + 23 -3-5V7 + 22-3-5-7 /3 + —

x = Xv.

1 5 1 (27T +l)3-2533T_

35(27T + 1)ïF(—, —,

12 12 2 (27T + 1)-95

254(1 +v/2)2T1 1

6

where

t! = 24 • 257 • 2441 + 2232 • 11 • 17923^2

+ 32V22 • 3 • 221047 937823 + 11 17 • 139 67672987V2,

and, finally, for x = 10/:

1 _5_ 1 (S24 + 231)3-26033S24_

H(-l + V5)(£24 + 231)i(12'12'2; (E24 + 23i)3

}-2T.3Ï-5E4

,where

Ë1:= 1 + V5 + V10 + 6V5.

Remark 52. The evaluation at t = 2z is that given by Beukers and Wolfart ([BW86]Theorem 3). The evaluation at r = 3z was found by Flach [F189]. Flach also

asserted a value for r = Ai, but this was found to be false by Joyce and Zucker

[JZ91]. For this, they used another method which also produced the evaluation at

x = 5i. Our evaluations at x = 6i, li, 8i, 10/ are new.

PSi

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if /

Ct iK.

_-*L W V4

153

Acknowledgments

My first thought goes to Prof. Eva Bayer-Fliickiger whose intervention at the begin¬ning made all of this possible. Her enthusiasm and dynamism were very encourag¬

ing.

I wish to express my sincere gratitude to Prof. Gisbert Wüstholz for having taken

me under his directorship and proposed to me an interesting subject which lies at

the confluence of algebra, number theory and geometry.

The sporadic meetings I had with Prof. Paula Cohen were always a great source

of motivation. I am also obliged to her for some pertinent remarks on previousversions of this work.

Many thanks to Prof. Horst Knörrer for some fruitful discussions and to Prof.

Richard Pink for his time and help in solving a problem.

Special thanks to my teaching assistant colleagues for many discussions, in particu¬lar to Walter Gubler, Mauro Triulzi and Thomas Loher. Thanks to Roberto Ferretti

for a useful reference.

The last thoughts are dedicated to my family for their patience and support and for

the encouragement given to me by the numerous mathematicians among them.

4*

eue Leer /

•in%

155

Curriculum Vitae

I was born in Geneva on the 11th of June 1973. In this town, I attended primaryand secondary school and obtained the Maturité Scientifique in June 1992 at the

Collège Calvin. In October 1992,1 began my studies in mathematics at the Univer¬

sity of Geneva. My diploma thesis "Puissance d'une Matrice carrée et Polynômesde Dickson de deuxième Espèce" was written under the direction of Prof. Michel

Kervaire and completed in March 1996. Having obtained the Diplôme de Mathé¬

maticien in July 1996, I moved to Zurich in September to begin a PhD under the

direction of Prof. Gisbert Wüstholz. From that time, I have been employed as a

teaching assistant at the Swiss Federal Institute of Technology of Zurich (ETHZ).

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In Copyright - Non-Commercial Use Permitted Rights ...23817/eth-23817-02.pdfAbstract Thisworkis devotedto hypergeometricseries. In the first chapter, weconstruct explicitely abelian