Noncritical Belyi Maps

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Noncritical Belyi Maps

Shinichi Mochizuki

May 2004

In the present paper, we present a slightly strengthened version of awell-known theorem of Belyi on the existence of Belyi maps. Roughly speaking,this strengthened version asserts that there exist Belyi maps which are unramifiedat [cf. Theorem 2.5] or even near [cf. Corollary 3.2] a prescribed finite set ofpoints.

Section 1: Introduction

Write C for the complex number field; Q C for the subfield of algebraicnumbers. Let X be a smooth, proper, connected algebraic curveover Q. IfF is afield, then we shall denote by P1F the projective lineoverF.

Definition 1.1. We shall refer to a dominant morphism [ofQ-schemes]

: X P1Q

as a Belyi map if is unramified over the open subscheme UP P1Q

given by the

complement of the points 0, 1, and ofP1Q

; in this case, we shall refer to

UXdef= 1(UP) X as a Belyi openofX.

In [1], it is shown that Xalways admitsat least one Belyi open. From this pointof view, the main result(Theorem 2.5) of the present paper has as an immediateformal consequence (pointed out to the author by A. Tamagawa) the followinginteresting [and representative] result:

Corollary 1.2. (Belyi Opens as a Zariski Base) If VX X is any opensubscheme of X containing a closed point x X, then there exists a Belyi openUX VX X such thatx UX . In particular, the Belyi opens ofX form a basefor the Zariski topology ofX.

Acknowledgements:

The author wishes to thank A. Tamagawa for helpful discussions during No-

vember 1999 concerning the proof of Theorem 2.5 given here.

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2 SHINICHI MOCHIZUKI

Section 2: The Main Result

We begin with some elementary lemmas:

Lemma 2.1. (Separating Properties of Belyi Maps) LetC R be suchthatC 2; let

S P1(Q)

be a finite set of rational points such that:

(i) 0, 1, S;

(ii) there exists anr Ssuch that0< r 1.

Suppose that Q\Ssatisfies the following condition:

(iv) / C, for all S\{0, }.

Writer= m/(m + n), wherem, n 1 are integers. Then the function

f(x) def= xm (x 1)n

satisfies the following properties:

(a) f({0, r, 1, }) {0, f(r), };

(b) f(x) = 0 (wherex C) impliesx {0, r, 1, } S;

(c) f() / f(S);

(d) (f() + f0)/(f() + f0) C for all S\{} such thatf() + f0 = 0.

Here, we writef0def

= min {f()}, where ranges over the elements ofS\{}.

Proof. Property (a) is immediate from the definitions. Property (b) follows im-mediately from the fact that:

f(x) =xm1 (x 1)n1 {(m + n)x m}

This computation also implies that for real x > 1, we have f(x) > 0, hence thatf(x) ismonotone increasing, for real x >1. In particular, since, by condition (iv), C 2 > , for all S\{0, }, we conclude that f()> f(), for all

S\{} such that >1.


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NONCRITICAL BELYI MAPS 3

Next, observe that since 1 S\{0, }, condition (iv) implies that C 2,sof()> 1. Since |f(x)| 1 for x [0, 1], we thus conclude thatf() / f(S), i.e.,that property (c) is satisfied.

Next, let us observe the following property:

(e) If S\{} satisfies > 1, then ( 1)/( 1) / 1;f()/f() (/)2 /.

[Indeed, as observed above, ; thus,f()/f() = (/)m {(1)/(1)}n (/)m+n (/)2 /.] Now we proceed to verify property (d) as follows:

Suppose thatn is even. Thenf() 0, for all S\{}, sof(0) = 0 impliesthat f0 = 0. Thus, if (S\{}) >1, then, by condition (iv) and property (e),we have: f()/f() / C, as desired. Since f(0) = f(1) = 0, to completethe proof of property (d) for n even, it suffices to observe that 0 < f(r) 1, so

f()/f(r) f() =m

( 1)n

C [since C 2, as observed above].Now suppose that n is odd. Thenf(x) 0 for x [0, 1], so [since f(x) = 0

for x (0, 1) x = r] we conclude that:

f0= |f(r)|= {m/(m + n)}m {n/(m + n)}n

Note, moreover, that this expression for f0 implies that 0< f0 1

4. [Indeed, this

is immediate in the following three cases: m, n 2; m= n = 1; one ofm, n is = 1and the other is 3. When one ofm,n is = 1 and the other is = 2, it follows fromthe fact that ( 1

3) (2

3)2 1

4.] Then if >1, then eitherf() f0, in which case

(f() + f0)/(f() + f0) f()/{2 f()} 1

2 (/)2 (/) C

[by property (e)] orf() f0, in which case

(f() + f0)/(f() + f0) f()/{2 f0} 2 f() = 2m( 1)n C

[since 0< f0 1

4, C 2]. On the other hand, if {0, 1}, then

(f() + f0)/(f() + f0) = (f() + f0)/f0 f() m ( 1)n C

[since C 2, as observed above]. This completes the proof of property (d).

Lemma 2.2. (Belyi Maps Noncritical at Prescribed Rational Points)Let

S P1(Q)

be a finite set of rational points such that:

(i) 0, S;

(ii) S\{0, } implies >0.


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Suppose that Q\Ssatisfies the following condition:

(iii) / 2, for all S\{0, }.

Then there exists a nonconstant polynomialf(x) Q[x] which defines a morphism

: P1Q P1Q

such that: (a) (S) {0, 1, }; (b) () / {0, 1, }; (c) is unramified overP1Q\{0, 1, }.

Proof. Indeed, we induct on the cardinality |S| ofSand apply Lemma 2.1 [with,say, C= 2] to the set S P1Q, for some appropriate positive rational number. Then, so long as |S| 4, the polynomial f(x) + f0 of Lemma 2.1 determinesa morphism P1Q P

1Q, unramified away from the image of S, that maps , S to

some , S that satisfy conditions (i), (ii), (iii) of the present Lemma 2.2, but forwhich the cardinalities ofS, S satisfy |S| < |S|. Thus, by applying the inductionhypothesis and composing the resulting morphisms P1Q P

1Q, we conclude the

existence of an f, as in the statement of the present Lemma 2.2.

Lemma 2.3. (Separation of Collections of Points) Let

S P1(C)

be a finite set of complex points. Then for any realC >0 and C\S P1(C)\S,

there exists a C such that the rational function

f(x) = 1/(x )

satisfiesf() = 0, ; f() = ; and|f()| C |f()|, for all S. Moreover,if Q, then one may take Q.

Proof. Indeed, it suffices to take such| | issufficiently small.

Lemma 2.4. (Reduction to the Rational Case) WriteQ C for the

subset of algebraic numbers. Let

S P1(Q) P1(C)

be a finite set ofQ-rational points. Suppose that Q\S. Then there exists anonconstant rational functionf(x) Q(x) which defines a morphism

: P1Q

P1Q

such that, for someS P1(Q), we have: (a) (S) S; (b) () P1(Q)\S;(c) is unramified overP1

Q\S. Moreover, ifS, are defined over a number field

F, then may be taken to be defined overF.


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Proof. First of all, we observe that by applying the automorphism x x ,we may assume that P1(Q). Moreover, under the hypothesis that P1(Q),we shall construct a f(x) satisfying the required conditions such that f(x) Q(x).Also, we may replace Sby the union of all Gal(Q/Q)-conjugates ofSand assume,without loss of generality, that S is Gal(Q/Q)-stable.

IfF is a finite extensionofQ, then let us refer to the number [F : Q] 1 asthe reduced degreeofF. Write

m(S)

for the maximum of the reduced degrees of the fields of definition of the variouspoints contained in Sand

d(S)

for the sum of those reduced degreesof the fields of definition of the various pointscontained in Swhich are equal to m(S). Thus,d(S) = 0 if and only ifm(S) = 0 ifand only ifS P1(Q).

Now we perform anested inductionon m(S),d(S): That is to say, we inducton m(S), and, for each fixed value ofm(S), we induct on d(S). Ifm(S), d(S) = 0,

then let0 S\P1(Q) be such thatd0def= [Q(0) : Q] is equal tom(S)+ 1. Then by

applying an automorphism (with rational coefficients!) as in Lemma 2.3 and thenmultiplying by some positive rational number, we may assume that || 1, for all S\{}, while|| C, for some sufficiently largeC, where sufficiently largeis relative to d0. Letf0(x) Q[x] be the monic minimal polynomialfor 0 over Q.Then one verifies immediately that all of the coefficients off0(x) have absolute value

dd00 . In particular, it follows that the value off0 at every S\{}, as well

as at every element of the set S0 of roots of the derivative f

0(x) has absolute valuebounded by somefixed expression ind0. Thus, for a suitable choice ofC, it follows

that f0() / S def= f0(S)

f0(S0). Moreover, sincef0(0) = 0; [Q(

) : Q]< d0 forevery f0(S0) [since f0(x), f0(x) Q[x]; f

0(x) has degree d0 1], it followsthat S is Gal(Q/Q)-stable and satisfies the property that either

m(S)< m(S)

orm(S) =m(S); d(S)< d(S)

thuscompleting the induction step. In particular, replacing SbyS

, byf0(),applying the induction hypothesis, and composing the resulting morphisms yieldsa morphism as in the statement of Lemma 2.4.

Theorem 2.5. (Belyi Maps Noncritical at Prescribed Points) Let X bea smooth, proper, connected curve overQ and

S, T X(Q)

finite sets ofQ-rational points such thatS

T = . Then there exists a morphism

: X P1

Q


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such that: (a) is unramified overP1Q

\{0, 1, }; (b)(S) {0, 1, }; (c) we have

(T)

{0, 1, } = . Moreover, ifX, S, andTare defined over a number fieldF,then may be taken to be defined overF.

Proof. Since X(Q) is infinite, we may always adjoin to T extra points of X(Q)that do not lie in S; in particular, we may assume, without loss of generality, thatT hascardinality 2gX+ 1, where gX is the genus ofX. Write

D def=

tT

[t]

for the effective divisoron Xobtained by taking the formal sum of the points inT, each with multiplicity one; denote the associated line bundleOX(D) by L andthe canonical bundleofXbyX . Also, we shall write s0 (X, L) for the section

[uniquely determined up to a Q

-multiple] whose zero divisor is D. Thus, thedegree deg(L) ofL is 2gX+ 1 1. In particular, ifx X(Q), then

deg(X L1(x)) (2gX 2) (2gX+ 1) + 1 = 2

so (X, X L1(x)) = 0. Since, by Serre duality, (X, X L1(x)) is dualto H1(X, L(x)), we thus conclude that H1(X, L(x)) = 0. Now if we considerthe long exact cohomology sequenceassociated to the exact sequence of coherentsheaves on X

0 L(x) L L k(x) 0

[where k(x) is the residue field ofX at x] we obtain an exact sequence

. . . (X, L) L k(x) H1(X, L(x)) . . .

i.e., we have asurjection(X, L) L k(x). SinceQis infinite, it thus followsthat there exists an s1 (X, L) such that s1(t) = 0 for all t T. Thus, thelinearseriesdetermined by the sectionss0,s1 ofL hasno basepoints, hence determines afinite morphism

: X P1Q

such that the pull-back by of the unique [up to isomorphism] line bundle of degree

1 on P1Q is isomorphic toL; maps everyt Tto the point 0 ofP1Q. Moreover,

since every point of the support ofD hasmultiplicity onein D, isunramifiedoverthe point 0 ofP1

Q; since no point ofSlies in the support ofD, this point 0 of

P1Q

does not lie in the set (S).

Thus, in summary, byreplacingXbyP1Q

, Tby the point 0 ofP1Q

, and Sby

the union of(S) and the points of P1Q

over which ramifies, we conclude that

we may reduce to the case X = P1Q

, T = {}, for some P1(Q)\{}. Next,

by applying Lemma 2.4, one reduces to the case X = P1Q

, S P1(Q), T = {},

for some P1

(Q)\{}. Finally, by applying an automorphism as in Lemma 2.3


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[for, say, C= 4], followed by a suitable automorphism of the form x x +,where {1} and Q, gives rise to a situation in which the hypotheses ofLemma 2.2 are valid. Thus, Theorem 2.5 follows from Lemma 2.2.

Section 3: Some Generalizations

Corollary 3.1. (Belyi Maps Noncritical at Arbitrary Sets of PrescribedCardinality)Let n 1 be an integer; X a smooth, proper, connected curve overQ and

S X(Q)

a finite set ofQ-rational points. Then there exists a finite collection of morphisms

1, . . . , N :X P1

Q

such that: (a) i is unramified overP1Q

\{0, 1, }, for alli = 1, . . . , N ; (b)i(S)

{0, 1, }, for all i = 1, . . . , N ; (c) for any subsetT X(Q) of cardinality n forwhichS

T= , there exists ani {1, . . . , N } such thati(T)

{0, 1, } = .

Proof. Note that we may think ofT as a Q-valued point of the n-fold product

Y def

= (X\S)n of (X\S) over Q. Then observe that for any : X P1Q

such that:

(a) is unramified over P1Q

\{0, 1, }; (b) (S) {0, 1, }, the subset

U Y(Q)

ofy Y(Q) for which (y)

{0, 1, } = [where, by abuse of notation, we write(y) for the subset of P1

Q(Q) which is the image under of the subset of X(Q)

determined by y] is nonempty and open [in the Zariski topology]. Moreover, byTheorem 2.5, the U coverY(Q) [i.e., as varies over those morphisms satisfyingthe conditions (a), (b)]. Since Y isquasi-compact, we thus conclude that there existfinitely many 1, . . . , N such that Y(Q) is covered by U1 , . . . , U N , as desired.

In the following, we shall refer to as a locally compact fieldany completion ofa number field at an archimedean or nonarchimedean place.

Corollary 3.2. (Belyi Maps Noncritical Near Arbitrary Points of Pre-scribed Degree) Let c, d 1 be integers; X a smooth, proper, connected curveover a number fieldF Q; V a finite set of valuations (archimedean or nonar-chimedean) ofF. Ifv V, then we denote byFv the completion ofF atv. Thenthere exists a finite collection ofmorphisms

1, . . . , N :X P1

Q


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and, for eachv V, a locally compact field Lv and acompact set

Hv (P1\{0, 1, })(Lv) P

1(Lv)

satisfying the following properties:

(i) F Fv Lv [i.e., Lv is a topological field extension ofFv];

(ii) Lv contains allQ-conjugates of all extensions ofF of degree d;

(iii) everyi (wherei {1, . . . , N }) is defined over everyLv (wherev V);

(iv) i is unramified overP1

Q\{0, 1, }, for alli= 1, . . . , N ;

(v) for any subsetT X(Q) of cardinalityc consisting of pointsx Twhose field of definition is of degree d over F, there exists an i {1, . . . , N } such thati(x) Hv, for allx T, Gal(Q/F), v V.

Proof. As in the proof of Corollary 3.1, write Y def

= Xn for the n-fold product

X over F, where we set n def= c d. Thus, for any T X(Q) as in the statement

of Corollary 3.2, (v), the conjugates over F of the various x T [in any order,with possible repetition] form a point Y(Q). Let Lv be a locally compact fieldcontainingFv , as well as all Q-conjugates of all extensions ofFof degree d. Thenobserve that for any : X P1

Qwhich is defined over all of the Lv [as v ranges

over the elements ofV] and unramified over P1Q

\{0, 1, }, the subset

U Y(LV)def=

vV

Y(Lv)

ofy Y(LV) for which (y)

{0, 1, } = [by abuse of notation, as in the proofof Corollary 3.1] is nonempty and open relative to the product topology of theZariski topologies on theY(Lv), hence a fortiori, relative to the product topologyof the topologies on the Y(Lv) determined by the Lv . Moreover, by arguing asin the proof of Corollary 3.1 using Theorem 2.5 and the Zariski topology, we mayassume that the Lv are sufficiently largethat [in fact, finitely many] such U cover

Y(LV). Now since eachUislocally compactand contains acountable dense subset,it follows that each U admits an exhaustive chain of open subsets

V,1 V,2 . . . U

[i.e.,

j V,j = U] such that the closure V,j in U of each V,j is compact.

On the other hand, since Y is proper, it follows that Y(LV) is compact. We thusconclude that there exist finitely many1, . . . , N such that Y(LV) is covered byV1,j1 ; . . . ; VN,jN , where [by abuse of notation, as in the proof of Corollary 3.1, wewrite]

i(Vi,ji) i(Vi,ji ) i(Ui)

vV

(P1\{0, 1, })(Lv)


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for i= 1, . . . , N . Thus, we may take Hvto be theimagein the factor (P1\{0, 1, })(Lv)

of theunionof the compact subsets i(Vi,ji) of the product

vV (P1\{0, 1, })(Lv).

References

[1] G. V. Belyi, On Galois extensions of a maximal cyclotomic field, Izv. Akad.Nauk SSSR Ser. Mat.43:2(1979), pp. 269-276; English transl. inMath. USSR-Izv.14 (1980), pp. 247-256.

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Noncritical Belyi Maps