Betti numbers of congruence groups

ISRAEL JOURNAL OF MATHEMATICS 88 (1994), 31-72

BETTI NUMBERS OF CONGRUENCE GROUPS

BY

P E T E R SARNAK

Department of Mathematics

Princeton University, Princeton, NJ 08544, USA

e-mail: [email protected]

AND

SCOT ADAMS


University of Minnesota, Minneapolis, MN 55455, USA

e-mail: [email protected]

AND

APPENDIX ON REPRESENTATIONS OF COMPACT p-ADIC GROUPS

Z E ' E V RUDNICK


Princeton University, Princeton, NJ 08544, USA e-mail: [email protected]

ABSTRACT

It is shown that , for certain congruence families of Galois coverings of a

manifold, the individual Betti numbers are polynomial periodic functions

of the level. Similar results are proved for the dimensions of other spaces

of automorphic forms.

Received March 1, 1990 and in revised form September 1, 1993

31

32 P. SARNAK ET AL. Isr. J. Math.

0. Pre face

This paper constitutes a revised version of the Australian Mathematical Soci-

ety reprint [Sa]. The revision concerns the proof of Theorem 1.2 for the general

unipotent case. Since the writing of [Sa], a number of related publications have

occurred. E. Hironaka [Hi2, Hi3] has used the results below to prove polynomial

periodicity for Hirzebruch and more general surfaces. The recent paper of Rup-

pert [Ru] is closely related to Proposition 1.6 below and gives moreover bounds

for torsion points.

1. I n t r o d u c t i o n

It is well known that the genus of the Fermat curve F (N) ; X N + yN _ _ 1, is

(N - 1) (N - 2)/2. This is easily seen by realizing F(N) as a Z / N Z • Z / N Z

branched covering of p: and applying Hurwitz 's formula. In this paper we give a

general result about families of manifolds which ensures that their Betti numbers

are polynomial periodic.

Let K be a finite simplicial complex and let F = ~r:(K) be its fundamental

group. As usual let /~ denote the universal covering of K. For F ~ ,3 F of finite

index we get a F /F I Galois covering K I = / ~ / F ~ of K. We are interested in the

q-dimensional Betti numbers flq(K ~) as F ~ varies over "congruence groups". To

define these congruence groups we assume that we have a homomorphism

(1.1) ~fl: F ont~ H ( Z ) ,

where H is a matrix algebraic group defined over Q and H(Z) denotes the Z -

points of H. Moreover, we assume that H has strong approximation [K]. For

each integer N > 1, let

(1.2) RN: H(Z) --* H(Z/NZ)

denote reduction mod N. The congruence groups F(N) are then defined by

(1.3) r ( N ) = ker (RN o

Let HN : = RN(H(Z)). Clearly r / r (N) _ HN and K(N) := [,:/r(N) is an HN

Galois covering of K ( I ) .

The two most interesting examples of congruence groups are the cases

Vol. 88, 1994 BETTI NUMBERS OF CONGRUENCE GROUPS 33

(C.i) The map ~ in (1.1) is an isomorphism in which case F(N) is the familiar

principal congruence subgroup of H(Z) of level N.

(C.ii) The group H is Abelian and hence the homomorphism ~ factors through

Hi(K, Z) ~- F/[F,F]. The most interesting case is that of H(Z) = Z ~, r being

ill(K). Some concrete examples of (ii) are the following:

(A) FERMAT CURVES. The Fermat curves F(N) may be uniformized as

�9 (N) \7-/, where

{ ( : b ) ( a ~ ) ( 1 0 ) } 9(1) = F(2) = d �9 SL(2, Z) ~ (mod 2)

c 0 1 ' �9 (g ) = k e r ( R N o ~ ) ,

where ~: F(2) ~ Z 2 is the projection on homology. RN: Z 2 ~ (Z/NZ) 2 is as

above, reduction mod N and ~/ is the upper half plane. For more details see

Lang ILl]. Actually to be more precise F(N) is the compactification (branched)

of O(N) \ 7-/. Note that while O(N) is a congruence group in our sense it is not

so according to the usual definition IS].

(B) LINKS IN S 3. Let L C S 3 be a p-component link; that is disjoint union of

tt circles. Let F = ~r1($3 \ L). It is well known and easily seen that HI(S3\ L) ~- Z u. Let ~: F --* Z u be the Abelianizer and RN as above. We get in this way for

each N a uniquely defined (Z/NZ) ~ covering M(N) of S a \ L. When L is a knot

(i.e., p = 1) these are the well studied cyclic covers [F 1].

(C) HIRZEBRUCH SURFACES [Hh]. Let t l , . . . , g r be an arrangement of lines

in p2(R) C p2(C). The Hirzebruch surface E(N) for N > 1 is the (Z/NZ) r-1

branched cover of p2(C) branched over g l , . . . , g~, see [Ha] for a discussion. Hirze-

bruch computed the characteristic numbers of E(N). Their Betti numbers for

certain t?l , . . . , g~ and N were computed by Ishida [I] and Hironaka [Ha].

Before turning to our results we consider a related problem. Let G be a

semisimple real, complex, or p-adic group. Let F < G be a cocompact lat-

tice. The right regular representation of G on L2(F\G) decomposes into ~ i ~ 1 Vi

where Vi are irreducible unitary representations of G. We denote by G the set

of equivalence classes of such representations. One of the fundamental problems

in automorphic form theory is to determine m(~r, F), the multiplicity with which

some ~r E G appears in the above decomposition. It is known [B-W] that for

certain r ' s this multiplicity corresponds to cohomology of F. We call such 7r's

34 P. SARNAK ET AL. Isr. J. Math .

cohomological. Now let F(N) < F be a congruence family in the sense defined

above. Our aim is to study m(Tr, F(N)) as a function of N.

Note that the group H N • G acts unitarily on L2(F(N) \ G) by

(1.4) (h, g) f ( x ) -* f ( h - l x g).

Decomposing this into irreducibles we get multiplicities m(p | 7r, F(N)) with

p E HN. For v _> 1 and integer, we define the v-dimensional part of m(Tr, F(N))

to be

~mOr, r( N) ) =

We can recover mQr, r (N)) by

m(p,r (N)) . dim p=v pE flN

oo

r(N)) -- Z r(N)) v = l

(the sum of course is finite).

Returning to the simplicial complex setting, we have HN acting as Galois

group of K ( N ) \ K ( 1 ) and hence HN acts on the eohomology groups Hq(K(N)) .

Decomposing this action into irreducible gives

H q ( K ( N ) ) : ( ~ W, . i

We define the v-dimensional part of the qth Betti number ~/3q(K(N)) to be

(1.5) ~ t 3 q ( K ( N ) ) = d i m ( d i m ~ w ~ = W i ) .

We observe that i f H is Abelian then 1/3q(K(N)) = ~q (K(N) ) and lm (Tr, F(N)) =

m(Tr, F) since all irreducibles of H are 1-dimensional.

Definition 1.1: A sequence a(n), n >_ 1, is polynomial periodic if there are

periodic bj (n) , j = 1 , . . . , q such that a(n) = ~ q = o b j ( n ) n j. |

We can now state the periodicity theorem

THEOREM 1.2: Assume that H is either unipotent or semisimple then

(i) I f r = 7rl(K) is as above and F(N), N _> 1, a congruence family, then

~I3q ( K ( N ) ) is polynomial periodic in N.

(ii) I f G is Archimedian and 7r is cohomological or if G is p adic and 7r is

arbitrary, then "m (Tr, F(N)) is polynomial periodic in N.


Remark 1.3: Since the general H is roughly a semidirect product of a unipotent

part with a semisimple part (recall that H has strong approximation) we expect

that the Theorem is valid for the general H. In lAd, Theorems 6.7 and 6.8], this

question is mostly settled in the affirmative; however, some additional hypotheses

are needed.

COROLLARY 1.4: Let L be a p-component link in S 3 and set M(N) to be the

(Z/NZ) ~ cover of S 3 branched along L; then ~31(M(N)) is polynomial periodic.

This Corollary in the case of knots is known and due to Goeritz [G] and Zariski

[Z] and to Sumners [Su] in higher dimensions. In fact, for knots one can show

that the sequence is periodic (i.e., q = 0 in Definition 1.1). In general this is not

so as the following example shows: The link L in Figure 1.5 has two components.

We can write down gl (M~(N)) compactly for all N by use of the generating

function ~-~'~N=I ~ 31 (M~(N))/NS" In Section 4 we show

-~ ~I(Mu(N)) (1.6) Ns - 2 (1 + 2 -8 + 6 -8) ~(s) + 2 . 6 - ~ ( ( s - 1),

N=I

where {(s) = ~-~N=I N-~" Here MY(N) in the unbranched (Z/N Z) 2 covering of

S3\ L.

Figure 1.5.

COROLLARY 1.5: Let E~( N) be the unbranched Hirzebruch surfaces over a fixed

configuration of lines; then the Betti numbers ~I( E~( N) ) are polynomial periodic

in N.

In [Ha], E. Hironaka developed a method for computing the difference between

the Betti' numbers of the branched and unbranched surface. Using this result,

she has proved periodicity in the branched covers E(N) (see [Ha2]), as well as

more general such surfaces [Ha3]. For cyclic coverings of p2, such results follow

from Libgober [Lr2].

Our Theorem does not apply to m(~r, F(N)) when G is Archimedean and 7r is

not cohomological. In these cases we only get asymptotic results. An analysis of


m(Tr, &(N)) for &(N) the Fermat groups and 7r �9 PSL(2, i t ) , with 7r principal or

complementary series will be given elsewhere ([P-S]). Suffice it to say here that

it still appears to be the case that m(~r, O(N)) is periodic in N, though a proof

of this seems out of reach at present.

There are a number of ingredients that go into the proof of Theorem 1.2.

In Section 2 we discuss the variety of representations of F into GL(u, C) and

the important algebraic subsets connected with the cohomology problem. In

Section 3 the problem of counting torsion points on algebraic subsets is studied.

Specifically let T be an r-dimensional real torus and let W C T be an algebraic

subset (i.e., the zero set of a trigonometric polynomial or intersection of such

sets). Denote by tor (T) the set of torsion points (i.e., elements of finite order in

T). The following was conjectured by Lang [L2]:

PROPOSITION 1.6: There are a finite number of rational planes 7rl,...,7c~

(a rational plane is a closed connected subgroup o f T or a translate thereof by a

torsion point) contained in W such that

(Ol) tor (T) n W = tor (T) n 7rj .

In Section 3 an elementary and algorithmic proof of this Proposition is given.

Given W it produces the planes 7r l , . . . , 7r~ which is crucial for the purposes of

computing the periodicities and polynomials. Another proof of this conjecture

is contained in the work of Laurant [Lt]. The importance of Proposition 1.6 lies

in the fact that as far as torsion goes we may linearize W, that is replace it by

planes. It is this that is responsible for the polynomial periodicity. In Section

4 we complete the proof of the Theorem and its Corollary. Sections 5-11 are

devoted to the proof of a technical fact (Theorem 11.3) which is crucial to the

proof of Theorem 1.2, when H is unipotent.

ACKNOWLEDGEMENT: The first author would like to thank A. Adem, P. Cohen,

D. Fried, A. Lubotzky and J.-P. Serre for illuminating discussions on various

aspects of this paper.

2. Representat ion varieties and cohomology

Let K be a finite simplicial complex/~ and F = 7rl(K) as in Section 1. If R is

a finite-dimensional unitary representation of F in GL(V), then the R-twisted


cochain complex C(/~, R) is defined by

(2.1) C q ( R , R ) = {F: Cq(k) --~ V, F is linear and F ( T a ) = R(7) F(a) for 7 e F}.

We have a coboundary operator

eq: cq(k, R) Cq+l(k, R)

given by

(2.2) (~qF) (a) = F(Oa).

The cohomology groups H*(/~, R) are defined in the usual way (here and else-

where Cq(/() is to be taken with complex coefficients). Our first problem is to

determine the behavior of flq(/~, R) = d imHq( / ( , R) as a function of R. While

the variety of all (up to equivalence) representations of F in GL(~, C) can be

quite complicated, the set of those which have finite image (which are all that we

need consider) is much more tractable. In fact as is shown in Lubotzky-Magid

[L-M], it follows from a theorem of Jordan, see [C-R], that all such R's which are

irrducible and of dimension ~ factor through a fixed quotient A of F. Moreover

A is Abelian by finite. Rudnick [R] gives a description of the variety V(A, v) of

all irreducible representations (up to equivalence) of such a A. In what follows

we will denote by T(B) the torus of 1-dimensional unitary characters of a group

B. Rudnick shows that there are finite index subgroups H 1 , . . . , IlL of A such

that

(2.3) V(ZX, L,) ~ U I n d ~ ( ~ ) . j = l XE j)

We are thus led to examine the functions

(2.4) Fj(X) = fl(~)(-~,Ind/~j(X)), X e T(Hj).

To do so we will use Hodge theory in the form of the finite combinatorial Lapla-

cian. For our purposes this has the advantage of dealing with the automorphic

form problem in the same way.

As in Ray-Singer [R-S] choose a preferred basis F?~,3 = alq | ej of Cq(f4, R) as

follows: Let el . . . . , em be an orthonormal basis of V and let Lr~,..., ~qq be the


set of q-simplices of K (which we think of as embedded in/s For i = 1 , . . . , uq,

j = 1 , . . . , m

{ 0 i f i Cg, (2.5) aq| R('y)ej i f i = L

Then 6 has a matrix representation relative to the bases b:q,,3 and F(~ +1)., �9 . Using

these bases we may define 5" and also the combinatorial Laplacian A~c): Cq(~:, R) --* Cq(K, R) by

(2.6) zx~c ) = (6(q+1)) * 6q + 6 q - i ( 6 q ) * .

As with the usual Hodge theory

(2.7) dimker ATe) = 13q(k, R).

Returning to our R's of the form (2.4), let r F ~ A be the projection and

denote by R(X) the representation of F given by

(2.8) R(X) = IndaH(X)o r

Let H 6 1 , H 6 2 , . . . , H 6 ~ , be coset representatives of H in A. An orthonormal

basis of V = V(Ind~/(X)) can be chosen in the form e l , . . . , e u when

ej(h6r) = ~ x(h) if j = t, (2.9)

L 0 otherwise.

PROPOSITION 2.1: The matrix of A~)(R(x)) relative to the basis a~ | ej of Cq( [s R(X) ) has entries which are trigonometric polynomials in ~.

The proof is a straightforward verification. We note for later that Proposition

2.1 holds equally well with R(X) replaced by

(2 .10) k ( x ) = n o | x ,

where R0 is a u-dimensional representation of F. For A E C and X E T(H) let

p(A, X) denote the characteristic polynomial

(2.11) p(A, X) = det(A - A~)(R(~))) .

Define the algebraic subsets of T(H) , Vo D 1/1 D 1/2.-. by

(2.12) 1 / I={X[p (0 , X ) = 0 } , Vk= X ~ ( O , x ) = O V1Vk_l.


In view of Proposition 2.1 the Vk's are indeed algebraic. Note that

(2.13) yk = {xl z (k, > k}.

Hence

1 if)~ c V, (2.14) /~q(/~, R(~)) = E e(~, Vk), where e(X, V) = 0 if ~ ~ V.

k_>l

The last sum is finite since Vk = 0 for large k.

Now recall that only the R(x) 's which are of finite image play a role in

our analysis. For these clearly )~ E T(H) must be a torsion point, that is

we are interested in ~q(/~, R(X)) for X ~ tor (T(H)). Applying the lineariza-

tion Proposition 1.6 we conclude that for each Vk there are planes r[k), . ,Tr (k) �9 . ~

and integers m~ k) m (k) (which come from inclusion-exclusion) such that for ' �9 " �9 ' ~ k

X e tor (T(H))

(2.15) e(X, Vk) = E m(k) E(X'r~))" p=l

Combining this with (2.14) we obtain our basic formula for Fk(X) in (2.4) (we

drop the index k).

PROPOSITION 2.2: There are planes r l , . . . , rL and integers ml , . . . ,mL such

that for X e tor (T(H))

L

Zq(k, = E m .

Thus far we have examined only the cohomological case, F = 7rl(K). To

deal with part (ii) of Theorem 1.2 when F is a lattice in a p-adic group G, one

proceeds along similar lines. Let m(Tr, F, R(X)) have the obvious meaning, where

X E T(H) as above. Now say G is rank 1 (actually it is only for rank 1 that

the variety of representation of F into GL(v, C) is not zero dimensional); then by

the duality theorem [G-G-P], m(~r, F, R(X)) may be realized as the dimension of

a certain eigenvalue of a vector valued Laplacian over the finite graph F\G/U,

U being a maximal compact subgroup of G. Hence the setup is identical to what

we have and the only change needed is that in (2.6) we are interested in the

general eigenvalue of A rather than just A = 0. Of course there was nothing

40 P. S A R N A K E T AL. Isr. J. Math .

special about A = 0 in what followed. One proceeds to derive an expression for

m ( r , F, R(X)) where X �9 tor (T(H)) , just like the one in Proposition 2.2.

We remark that in the Archimedean case F \ G / U is a compact manifold but

unfortunately det(A(R(x))) is no longer a trigonometric polynomial in ;~. One

can show that it is real analytic in X but we will not pursue this here.

3. To r s ion po in t s on var ie t i es a n d l i nea r i za t ion

We turn to the proof of Proposition 1.6. The approach below is simpler than our

original and was suggested by Paul Cohen.

LEMMA 3.1: Let al , . �9 � 9 aR be nonzero complex numbers; then there is a number

M = M ( a l , . . . , aR) such that any solution in roots of unity 61, . . . , en of

R

(3.1 ) >__: = 0 j = l

satisfies

(3.1)' (Ej ~-1)~ = 1

for some j # k and some 1 < v < M.

Proof." We begin by assuming that C~l,..., a n E Q, in which case we show M

may be chosen to be ( R + 1) n+l. Let e l , . . . , en be a solution of (3.1) and assume

the conclusion of (3.1)' fails. Let n be the 1.c.m. of the orders of r en. If " e l 8 2 e k the factorization of n is Pl P2 "-'Pk then we may write

�9 v(1) . v(2) (3.2) cj = (epiC) J (r . . . (r

where r = e(1 /m) = exp(2~ri/m). Now if some pj, say Pl, is larger than R + 1,

then we proceed as follows:

There is some ej for which (u~l ) ,p l )= 1. We may write (3.1)as

R

j = l

~kj where r/i~';;l_~ ~p~l

and (v l ,p l ) = 1.

- - _ _ e 2 e 3 = ~ j , 0 < /Jj < P l -- 1, ~/j i s a P2 P3 . . . P ~ r o o t o f 1

Thus (3.3) gives an equation for ~p~l over the field K =


q (r (p ;2 , . . . , (p;k) of degree at most pl - 1. The extension K(~p~ ) over

K has degree Pl or Pl - 1 and, since R < Pl - 1, (3.3) is impossible (since then

(3.3) is not the cyclotomic equation) unless the equation trivializes. In this case

one can proceed inductively since the power e I has been reduced and the side

conditions of the Lemma persist.

We may therefore assume that pj _< R + 1 for each j . We write (3.1) as

(3.4) E 3e T1 + - - + . . . + --0, j=l p~2 ~ ]

where now

+ + ]

For r = (r~, r 2 , . . . , rk) with pj ](rj we obtain from (3.2) and the Galois action of

Q (~n)/Q Z ~ j e ( r l m ~ j) rkm~J)~ j=l ~k P~ -[-' ' '~- P-~ ] =0"

Multiplying this equation by e ~a + . . . + ~ and summing over rj Pl P~ ] mod p~', pj ](rj, we get

(3.5) R

j=2 rj mod p~3

e (rl (m~J) e-p11 m~ 1)) rk (rn~ j) - rn~l)) ) + . . . + pekk

where f j ej or ej 1. Let m(J) Lmlr (;)p2 (~) "P;~] + . . . + [p~l .. _~-l__(J)] ~_ - - z �9 . t ) k _ l i i t k ]

so that ej = e(m(J)/n). If for a fixed j in the above type of sum m(j ) - m(, 1) are

all divisible by p ~ , then m ( J ) - m (1) mod ( ~ ) and hence

(~1~;1) plp2"''p~ ---- 1.

This is contrary to our assumption since PIP2. . .Pk < (R + 1) R+I. Hence for

each j at least one of m (j) - m (1) is not divisible by p[~. It follows that all the

series in (3.5) vanish and hence a l = 0. This completes the proof for the case of

most interest to us viz. a j E Q. In general one argues in a similar case with Q

replaced by the finitely generated field K = Q ( a l , . . . , a t ) and the Galois group

of Q ((n) replaced by that of K((n) over K. We leave the details to the reader.


We proceed now with the proof of Proposition 1.6. The proof is by induction

on the dimension of the torus T. We are allowing here a torus to be of the

form T = (R /Z) k • A, where A is a finite Abelian group. In such a case T is of

dimension k and the algebraic sets are algebraic sets at each connected component

of T. When dim T = 1 the result is clear since in this case a connected component

of T is either all of the set W, or W meets the component in a finite set. In the

general case, if d imT = r and W is given by equations

p l ( t ) = p 2 ( t ) . . . . . p . , ( t ) = o

on a connected component of T, then either these equations are all redundant

and W contains this connected component which is then one of the ~'s, or say

pl(t) is not trivially true. Say

(3.6) pl(t) = E a;~ )~(t), ~EF

where F is a finite subset of the dual group T*. The sets

(3.7) = { t l 1}

for A ~ #, A, p E F, and 1 < u < M(a:~), where M is the constant from Lemma

3.1, are all finite unions of rational planes of dimension r - 1. Hence since the

restriction of an algebraic set to a plane is algebraic, we conclude by induction

that the torsion points of TMW contained in the union of the finite set of Hx,~,~'s

are all on a finite number of rational planes in W. On the other hand Lemma

3.1 asserts that there are no torsion points of T M W outside of this union. This

completes the proof of Proposition 1.6.

We note that the above proof is effective and produces the planes inductively.

An example of this procedure is carried out in the next Section.

4. P r o o f of the T h e o r e m

We now complete the proof of Theorem 1.2 part (i). Part (ii) is proved similarly.

Fix a positive integer q and, as in the introduction, let RN: H(Z) ---* H(Z/NZ) denote reduction mod N. From our setup HN := RN(H(Z)) is the Galois group

of the covering R/F(N) over/~/r(1). Hence we may decompose Cq(R/r(N)) into invariant subspaces according to the irreducible representations of HN. Since


the action of HN on fibers of the covering is simple and transitive, it follows from

the decomposition of the regular representation that

(4.1) /3q([f/F(N)) = E d(R) t3q([f,R), RE ~I N

where d(R) = dim R. Hence

(4.2) "zq ( R / r ( g ) ) = u zq(k, R). REf/N

dim R=v

These R's are irreducible u-dimensional and of course finite representations of

F--this explains the assumption made in Section 2. To use the formula for

~q(/(, R(X)) developed in Proposition 2.2 we need to know which X's come up in

(4.2).

Assume H is unipotent, then H(Z) is a finitely generated nilpotent (discrete)

group. In this case, the representation variety V(A, u) in (2.3) takes an even

simpler form (see Lubotzky-Magid [L-M])

L

(4.s) v(A,u) = U U Rj | x, j = l xeT(H(Z))

where Rj is an irreducible (finite) representation of H(Z) of dimension v. More-

over, the representations in different classes Rj | T(H(Z)) and Rk @ T(H(Z)) for

k ~ j are inequivalent. It is possible that Rj | X' ~- Rj | X for some X r X' and

in this case {X: Rj | X ~- Rj} is a finite subgroup of T(H(Z)). For simplicity

we assume this group is {1}--the modifications needed to deal with the general

case dividing by this finite group are straightforward.

Let B(N,j) denote the set of all characters X �9 T(H(Z)) such that Rj | X factors to HN, i.e., such that the kernel of R 5 | X contains H(N). By equation

(4.2) and the above description of the representation theory of H(Z), we have

L

q(k, nj| j = l x e B ( N , j )

For all j , choose rational planes ~rl, . . . , 7r b and integers m l , . . . , m b as in Propo-

sition 2.2. Then

L lj

"~q( R /F( N) ) = u E E mulB( n'J) n 7rul, j = l p. : l


where {S{ denotes the cardinality of a set S.

It therefore suffices to show, for all j and #, that {B(N, j) n %{ is polynomial

periodic in N. In the case that H(Z) = Z ~ as in (C.ii) of the introduction, this

polynomial periodicity is easily seen. For the general case, we proceed as follows.

Fix j and tt; for all integers N > 0, define SN :---- B(N, j) n ~r,. We wish to show

that {SN{ is polynomial periodic in N.

Let Af be the collection of integers N > 0 such that, for some X E T(H(Z)),

we have ker(Rj | X) _D H(N). Let g l := min Af. By [Ad, Lemma 5.6] (which

depends only on strong approximation for H at the infinite place, and not on

any of the results in this paper), we find, for all integers N > 0, that: N1 {N iff

N E Af. In particular, for all integers N > 0 such that N1 XN, we have SN = ~,

so {SN{ = O. Therefore, it suffices to show that [SN1N{ is polynomial periodic in

N.

Choose X1 C T(H(Z)) such that ker(Rj | X1) D_ H(N1). Then, for all integers

N > 0, we have SN1N = )CI{X �9 xl lTr/~ [ ker(x) _D H(NIN)} . Fix any Xo �9

XllTrt, and define ~r := XolX-~l~r~. For all integers N > 0, define S~v := {X �9

~r { ker()coX) _D H(N)}. Then, for all integers g > 0, we have SN1N = Xox1S'N,N,

SO {SN1N{ = {SIN1N[. By Theorem 10.1, we see that {S~v { is polynomial periodic I in N, so {SN1N{ is as well, concluding the proof.

This completes the proof if H is unipotent. If H is semisimple then the above

analysis gives the result (but without any of the analysis of torsion points in tori)

because of the following result, a proof of which is given by Z. Rudnick in the

Appendix. (Also see [Ad, Theorem 5.12].)

PROPOSITION 4.1: Let H be a semisimple group over Qp. Let Zp denote the

p-adic integers. There are only tinitely many irreducible representations of the

compact group H(Zp) in any given dimension.

With this one finds that there are only a finite number of R's (independent of

N but depending on u) that come into play in (4.2). As a result the Theorem is

easy to establish.

The Corollaries for unbranched covers follow directly from the Theorem since

H in these cases is Abelian and hence we need only take u = 1 to get the complete

picture. For computational purposes say with links in S 3 we relate the varieties

1/1, V2,... of Section 2 to Alexander ideals IF-2]. In fact let L be a p-component

link; we can think of T(F) ---- (S1) it, where F -- 7r1($3 \ L), as sitting in (C*) ~.


The following Proposition follows easily from Fox [F-2], see also [M-M] and

Libgober [Lr].

PROPOSITION 4.2: Let W1 D W2 D W3" . . be the zero sets in (C*) ~ of the

elementary Alexander ideals El , E2, . . �9 of the link L. Then

Wk n T = Vk f o r k e r #

and (W, n T) u {1} = V~.

Here the Vk's are the sets defined in (2.12) for ~31 and H(Z) = Z ~ = Hi(F , Z).

The point is that the Alexander ideals are easy to compute. Thus for the link L

in Figure 1.5 one finds (see IF-T])

E1 : ( t l - l , t 2 - 1 ) ( 1 - t l + t ~ ) ( 1 - 2 Q - t 2 t 2 ( 2 - t l ) )

E2 : (1 -- t l -t- t l 2, (1 fl- t l ) ( 1 -F t2))

E3 : (1)

Hence using linear torus variables tj -~ e(0j) and Proposition 4.2 we have

V1 = {(0,0)} U { 1 - e(01) + e(20,) -- 0}

U{1 - 2 e(01) - e(2 8 1 ) e ( ~ 2 ) ( 2 -- e(01) ) ---- 0}, 1) v3 = 9.

Thus V3 and V2 are already linear. We apply the method of Section 3 to linearize

V1 and find two 1-dimensional rational planes viz.

~1 = {01 = 1/6}, ~2 = {01 = 5/6}

and three zero-dimensional planes

~3 = { ( 0 , 1 / 2 ) } , ~4 = { (1 /2 , 0)}, ~5 = {(0, 0 ) } .

Using this, one obtains (1.6) after a little calculation.

Concerning Corollary 1.4 for branched coverings, one uses instead of the

Alexander polynomial of L the reduced Alexander polynomial as described in

Mayberry and Murasugi [M-M]. Other than that the techniques apply directly

and yield Corollary 1.4. The rest of the paper (Sections 5 to 9) is concerned


with establishing the polynomial periodicity of the sequence [S~v [ described in

the proof of Theorem 1.2.

5. A division a l g o r i t h m for polynomial periodic functions

Let N := {1, 2, 3 , . . .} . For all s E N, define N8 := {1 , . . . , s}. For all s, t E Z

satisfying s _~ t, we define Zt8 := { s , . . . , t ~ . A function a: N ~ Z is said to be

p o s i t i v e if a(N) > O, for all N E N.

Definition 5.1: Let a: N --* Z be any function, let m E N and s E Z. We

say that a is a p o l y n o m i a l a long m Z + s if there exists a polynomial function

u: R ~ R such that u and a agree on (mZ + s) M N. If, in addition, the function

u satisfies u(0) -- 0, then we say that a is a 0 -van i sh ing p o l y n o m i a l a l o n g

m Z + s .

Definition 5.2: Let a: N --, Z. We define MOD(a) to be the set of all m E N

such that , for all s E Nm, the function a is a polynomial along m Z + s. We define

MODo(a) to be the set of all m E N such that , for all s E Nm, the function a is

a 0-vanishing polynomial along m Z + s.

Definition 5.3: Let a: N --* Z be any function. We say that a is p o l y n o m i a l

p e r i o d i c , or p .p . , if MOD(a) ~ ~. We say that a is 0 -van i sh ing p o l y n o m i a l

p e r i o d i c , or 0 -v .p .p . , if MODo(a) ~ 0. I

If a: N --, Z is 0-v.p.p., then MOD(a) = MOD0(a).

Definition 5.4: Let a: N --, Z be polynomial periodic and not identically zero.

Fix any m E MOD(a) and choose, for each s E Nm, the polynomial function

us: I t --~ t t such that a = us on (mZ + s) N N. Let S denote the collection of

s E Nm such that u8 is not identically zero. For each s E S, let uts: I t ---, I t

be defined by ur~(t) := us(mr + s). For s E S, let d~ := deg(ur~) = deg(us)

r We define the d e g r e e of a to and let c~ denote the leading coefficient of us.

be deg(a) := max{d~ Is E S}. Let R := {s E S[d~ = deg(a)}. We define the

full m t h se t o f l ead ing coef f ic ien ts and the m t h se t o f s ign i f ican t l e ad ing

coef f ic ien ts to be, respectively,

FLCm(a) := {c~ Is E S}, SLCm(a) := {cr [r E R}.

We will say that a is n o n v a n i s h i n g o n a r i t h m e t i c s e q u e n c e s if S = Nm.


LEMMA 5.5: Let a: N --* Z be p.p. Then there exists mo E N such that, for all

n E N, we have: mon E MOD(a) and FLCmon(a) C Z.

Proof: Let d := deg(a). Choose m E MOD(a). Choose, for each s E Nm, the

polynomial function us: R --* R such that a = u8 on (mZ + s) c3 N. For each

s E S, let uls: R --* R be defined by uls(t) := us(mt + s).

For each i E Z0 d, let Si denote the set of s E Nm such that deg(us) = i and let

Ci denote the set of leading coefficients of the polynomials in {u~s I s E Si}.

Then Co C_ Z since a (N) c_ Z. By [Jac, Proposition 7.26, p. 444], CIU. . "UCd E

Q. Choose an integer no such that no(C1 U . . . U Cd) C_ Z.

For all n E N,

FLCmn(a) = Co U nC1 U n2C2 U . . . U ndCd,

so we conclude the proof by setting mo := mno. I

Det~nition 5.6: If S, T C_ Z, and if 0 ~ S, then we use the notation SIT to mean:

sit, for all s E S, t E T, i.e., every element of S divides evenly into every element

of T. We make the convention that SI0 , for all S C_ Z\{0}.

LEMMA 5.7: Let a, b: N --* Z be p.p. and assume that b is not identically zero.

Assume that deg(a) > deg(b). Then there exists m E MOD(a) M MOD(b) such

that

FLCm(a) U FLC,~(b) C_ Z and FLCm(b)ISLCm(a ).

Proof: By Lemma 5.5, choose mo E N such that, for all n E N, we have:

mon E MOD(a) M MOD(b) and FLCmo,~(a) U FLCmo~(b) C_ Z.

Let no := 11 FLCmo(b), i.e., let no denote the product of all the elements of the

full moth set of leading coefficients of b.

Let c := deg(a) and d := deg(b), so, by assumption, d < c. Now, for all n E N,

we have

FLC,~on(b)indFLCmo(b) and SLCmo,~(a) = ncSLCmo(a),

so we conclude the proof by defining m := mono. I

48 P. SARNAK E T AL. Isr. J. Math.

LEMMA 5.8: Let a, b: N --~ Z be p.p. and assume that b is nonvanishing on

arithmetic sequences. Let m �9 MOD(a) M MOD(b) and assume that

FLCm(a) U FLCm(b) C_ Z and FLCm(b)ISLCm(a).

Then there exist p.p. functions qo, a0: N --~ Z such that deg(ao) < deg(a) and

such that a = bqo + co.

Proof'. Fix s E Nm. Choose po lynomia l funct ions us, vs: R ~ R such t ha t

" R ~ R b y u~s(t) := u s ( m r + s ) us = a and vs = b on ( m Z + s) fq N . Define uls, v s.

and v~s(t) := vs(mt + s).

Let R := {s �9 Nm [ deg(us) = deg(a)} . For each s �9 R, since the leading

coefficient of v~s divides the leading coefficient of u~, we may choose po lynomia l

0. R ~ 1~ such t ha t funct ions ws, u~.

I VlsWs _[_ 0 deg(u ~ < deg(us) and u s = u s

and such t h a t the coefficients of ws and of u ~ are all integers. For all s E N m \ R , 0 i define ws := 0 and u s := u s.

Define q0, a0: N --* Z by the rule: for all s E Nm, for all N C N U {0},

qo(mN + s) = ws(N) and ao(mN + s) = u~ |

We can now give an analogue for the division a lgo r i thm for po lynomia l per iod ic

functions. Notice t ha t the degree of the r ema inde r may equal the degree of the

divisor, so it is not possible to ob ta in an Euc l idean a lgor i thm in this case.

PROPOSITION 5.9: Let a, b: N --* Z be p.p. and assume that b is nonvanishing

on arithmetic sequences. Then there exist p.p. functions q, r: N --* Z such that

deg( r ) < deg(b) and such that a = bq + r.

Proof: The proof is by induct ion on deg(a) . If deg(a) _< deg(b), then we set

q := 0 and r := a; we therefore assume deg(a) > deg(b). Choose m as in L e m m a

5.7. Choose qo and ao as in L e m m a 5.8. By induct ion, there exist p.p. funct ions

q l , r l : N --* Z such t ha t d e g ( r l ) < deg(b) and such t ha t ao = bql + rl. Let

q := q0 + q l and let r := r l . I

In the case where b o t h divisor and d iv idend are 0-v.p.p, we can conclude t h a t

the r ema inde r is as well:


PROPOSITION 5.10: Let a, b: N --* Z be O-v.p.p., let q, r: N ~ Z be p.p. Assume

that a = bq + r. Then r is O-v.p.p.

Proof'. By Lemma 5.5, choose m �9 MOD(a) N MOD(b) V/MOD(q) M MOD(r ) .

For each s �9 Nm, choose polynomial functions us, vs, Ys, zs: t t ~ R such that ,

on (mZ + s) N N, we have us = a, vs = b, Ys = q and zs = r.

Fix s �9 Nm; we wish to show tha t zs(0) = 0. However, we know tha t us(0) =

vs(0) = 0 and that us = vsys + zs, so the result is immediate. I

6. F u r t h e r g e n e r a l i t i e s o n p o l y n o m i a l p e r i o d i c f u n c t i o n s

We will say tha t a function N --* Z is l i n e a r p e r i o d i c , or 1.p. if it is polyno-

mial periodic of degree _< 1, or if it is identically zero. A function N --* Z is

h o m o g e n e o u s l i n e a r p e r i o d i c or h . l .p , if it is 0-v.p.p. and linear periodic.

If t �9 N and ( X l , . . . , xt) �9 zt\{(0,..., 0)} then we define

g c d ( x l , . . . , x t ) := max{u �9 N i x 1 , . . . ,xt �9 uZ}.

LEMMA 6.1: Assume a,b: N --* Z are h.l.p. Assume further that a (N) r 0, for

all N E N . Then

N ~ gcd(a(N) , b(N)): N ~ Z

is positive and h.l.p.

Proof." By definition of gcd, for all N E N, we have gcd(a (N) , b(N)) > O.

Let Y C N be an ar i thmetic sequence on which there exist s, t, v C Z\{0} and

u C Z such that , for all N E Y, we have

a(N) = s N / t , b(N) = u n i v .

We wish to show tha t there exist w E Z and x C Z\{0} such that , for all N E Y,

we have

gcd(a(N) , b(N)) = w N / x .

Now, for all N C Y, we know tha t

tvgcd(a( N ), b( N ) ) = gcd( tva( N ), tvb( N ) ) = gcd( svN, t u N ) = [gcd(sv, tu) ]N.

So let w := gcd(sv, tu) and x := tv. I

50

COROLLARY 6.2:

bl (N) # 0, for all N C N. Then

g ~-* g c d ( b l ( g ) , . . . , bt(Y)): N --* Z

is positive and h.l.p.

LEMMA 6.3:

is periodic.

P. SARNAK ET AL. Isr. J. Math.

Assume that b l , . . . , bt: N --~ Z are h.l.p, functions and that

Suppose that a: N ~ Z is h.l.p. Then

N ~-~ min{u E N lua(N ) E NZ}: N --* Z

Proo~ Since, for all N E N and all a C Z, we have

N Ugcd( N, Nlua ~ gcd(N, a) a)

it follows that

N u~

gcd(N, a)

N

a(N)a ' (N) = bl(N)b'l(N ) + . . . + bt(N)b't(N ).

LEMMA 6.5: Suppose that a, b l , . . . ,b t : N ~ Z are all h.l.p, and that bt is

positive. For all N E N, let

a '(N) := min{N C N lua(N ) E bl(N)Z + . . . + bt(N)Z}.

Then a t : N ---* Z is periodic and there exist periodic functions b~,. . . , Ut: N --* Z

such that, for all N E N, we have

is h.l.p.

N H min{u e N lua(N ) C NZ}: N--* Z

min{u e N lua(N ) e N Z } - gcd(N, a(N))"

By Lemma 6.1, the gcd of a positive h.l.p, function and an h.l.p, function is

positive h.l.p.; it follows the the denominator above is positive h.l.p. Finally,

we note that the quotient of an h.l.p, function by a positive h.l.p, function is

periodic. |

The next lemma has a proof very similar to the last; we omit it.

LEMMA 6.4: I ra : N ~ Z is periodic, then


Proof: For all N �9 N, define

g(N) := gcd(b l (N) , . . . , b t (N) ) , c(N) := N/g(N) .

Then, by Corollary 6.2, 9: N ~ Z is positive, h.l.p, and so c: N ~ Z is periodic.

For all u, N �9 N, we have

ua(N) �9 b l ( g ) z + . . . + bt(N)Z N

r ua(N) �9 c - - ~ Z

ua(N)c(N) �9 NZ.

Since N ~ a(N)c(N): N --* Z is h.l.p., it follows from Lemma 6.3 that a~: N --* Z

is periodic.

By definition of a', for all N e N, we have g(N)[a(N)a'(N).

Let a", b~, . . . , b~: N ~ Z be defined by

a" := aa ' lg , b;' := b'/ := b /g.

Then a", b~',..., b~': N --+ Z are periodic.

For all N E N, there exist X l , . . . , xt C Z such that

(*) a"(N) = b~'(N)xl + . . . + b't'(g)xt.

Since the coefficients of (*) are periodic in N, we may solve ( ,) for finitely

many values of N, then repeat these solutions periodically in N. So there exist

b~, . . . , b~: N --+ Z periodic such that, for all N �9 N,

a" ( g ) = b~ ( g)b~ ( Y ) + . . . + b~t' ( Y)b~ ( g ) .

Multiplying this equation by g(N), we obtain the desired conclusion. |

LEMMA 6.6: Let a, b l , . . . , bt: N ---* Z be h.l.p., and let c: N ---* Z be periodic.

Assume, for all N E N, that c(N) ~ O. For ali N �9 N, define

a ' (N) := min{u �9 N [ u a ( N ) �9 b l ( g ) z + . . . + b t ( g ) z + c(N)Z}.

Then a': N ~ Z is periodic and there exist periodic functions b~l , �9 . . , b~'t. N ~ Z

and an h.l.p, function c~: N ~ Z such that, t'or all N �9 N,

a(N)a'(N) = bl(N)b~l(N) + . . . + bt(N)b~t(N) + c(N)c'(N).


Proof." By Proposition 5.9, there exist periodic functions a", b~', . �9 b"'t. N ~ Z

such that, for all N E N, a ( N ) -- a " ( N ) mod c ( N ) and

b l ( Y ) - b~(g) mod c (N) , . . . , b t (N) = b~t '(g)mod c (N) .

For all N E N,

a ' ( N ) = min{u �9 N i u a " ( g ) �9 b~(N)Z + . . . + b~t'(g)z + c(N)Z}.

Since a",by , �9 " �9 �9 bt, c. N ~ Z are periodic, it follows that a': N ~ Z is as well.

For all N �9 N, there exist X l , . . . , x t �9 Z such that

a " ( g ) a ' ( N ) �9 b~l~(N)Xl + . . . + b~(N)x t + c(N)Z.

Since a", a', by , . . b" �9 b I" �9 , t , c: N ~ Z are periodic, we may choose periodic b ~ , . . , r

N ~ Z such that, for all N �9 N, we have

I t l I t l a " ( N ) a ' ( N ) �9 bl ( N ) b l ( N ) + . . . + b t ( N ) b t ( N ) + c(N)Z.

Then, for all N �9 N, we have

a ( g ) a ' ( N ) �9 bl(N)b~l(N) + . . . + b t (g )b ' t (N ) + c ( g ) z .

Define c': N --~ Z by ca' - blbll . . . . . btb~

C t :__--

C

Then c' is the quotient of an h.l.p, function by a nonvanishing periodic function

and is therefore h.l.p. I

7. Generators for the integer points

Fix a positve integer n and a Q-subgroup H _< SLn.

We assume that H is a subgroup of the upper triangular unipotent matrices�9

(In w we will show that it is only necessary to assume that H is unipotent; the

element which conjugates such a group into the upper triangular unipotents may

be chosen to be a Z-point.) We assume that H is not the trivial (one-element)

group.

For each i , j E N,~, let pij: SLn --~ A 1 be the map which associates to each

matrix in SL~ its (i, j)-entry, which lies in affine 1-space.


Define m := n(n - 1)/2 and let 7r l , . . . , 7rm denote, respectively, the maps

pl21H, P231H , . . . , pn-2,n-l[H, Pn-l,n[H,

p13[H, p24[H . . . . . Pn-2,n[H,

pl,n-l[H, P2n[H pln[H.

Let F1 := H. If rl[F1 is not identically zero, then we set so := 0. Otherwise, we

define so := m a x { s e Z ~ I1rtlF1 . . . . . 7rs[F1 = 0}. Note that so < m, since

F1 = H ~ {e}.

Define r l := so + 1 and

F2:={heFll. l(h)=O), sl := max{s e Zm [' lIF2 . . . . . . . IF2=0} .

If Sl < m, then we define r2 :-- Sl + 1 and

F3:={heF21~r~2(h )=O} , s2:=max{seZm[Tv~21F3~2 . . . . . 7r~lF3=0}"

Continue until, at some l E N, we have sl = rn. Define Ft+l = {e}.

For all i E Nt, define S~ := Z ~

LEMMA 7.1: For ali i E Nt, the map Pi := ~r~]Fi(Z) is a homomorphism from

Fi(Z) onto an infinite cyclic group. The kernel of pi is Fi+I(Z).

Proof Fix i E Nt. One verifies from matr ix multiplication that pi is, in fact a

homomorphism. It is clear that pi(Fi(Z)) C_ Z and that ker(pi) ---- Fi+I(Z). We

must show that pi(Fi(Z)) ~ {0}.

Let V denote the (algebraic) Lie algebra of Fi; it is a subalgebra of the Lie

algebra of SLy. Let F denote a deep enough congruence subgroup of V(Z) that

exp(F) C_ Fi(Z). Since F is Zariski dense in V, it follows that Fi(Z) is Zariski

dense in exp(V) = Fi.

Then p~(F~(Z)) must be Zariski dense in p~(F~). It follows from the definition

of p~ and Fi that pi(Fi) is not the trivial group. Consequently, p~(FI(Z)) r {0},

as desired. |

For each i E Nl, let Ai be a preimage in Fi(Z) of a positive generator for

pi(Fi(Z)); see Lemma 7.1. For each i, note that pl(Ai) . . . . . pi-l(A~) = O,

while pi(Ai) > O.


LEMMA 7.2: For all h E H(Z), there exists a unique (n l , . . . ,nz) E Z l such that

h = A~ ~ . . . A ~ ' .

Proof: By definition of A1, there exists nl E Z such that A-[nlh E F2(Z). By

definition of A2, there exists n2 E Z such that A2n2(A-~'~h) E F3(Z). Con-

tinuing, we find n l , . . . , n t E Z such that A[ -n~ . . .A '~n~h E Ft+I(Z) -- {e}. So

h = A~ ~ . . . A~ ~, proving existence.

Now suppose A ~ ~ . . . A'~ ~ = A~ ~ . . . A~'; we wish to show that mi = nl, for all

i E N t .

By Lemma 7.1,

m l p l ( A 1 ) --- p l ( A ~ ~ . . . A ~ ~) = p l (A~ ' . . . A~' ) = n lP l (A1) ,

SO m l = nl . Then A'~ 2 . . . A'~ ~ = A~ 2 . . . A~' , and we continue using P2. I

The same proof shows

COROLLARY 7.3: For all i E N t , for all h E Fi(Z), there exists a unique

( n i , . . . , nt) E Z l-i-kl such that h = A ~ ' . . . A~' . I

8. G e n e r a t o r s for t h e N t h congruence subgroup

Let H, l and m be as in w For all k E Nm, let 7rk be as in w For each i E Nt

let Fi, ri, si, Si, Ai and Pi be as in w

Detlnition 8.1: For all i E Nt, for all k E Z m for all N E N, let Fi(k, N ) denote ?'i '

the set of all h E Fi(Z) such that

~r~, (h), 7r~,+l(h), . . . , 7rk(h) E NZ.

Definition 8.2: For all i E Nt, let 9ri denote the set of all maps B: N -* Fi(Z)

such that ~rr~ o B is h.l.p, and positive and such that 7rr~+~ o B , . . . , 7rr~ o B are all

0-v.p.p.

For all v E N, for all M 1 , . . . , M, E SLn(Z), define

( ( M 1 , . . . , M v ) ) := {M~ ' l . . . M ~ € E Z};

in general this is a set, not a subgroup.


LEMMA 8.3:

all N E N,

Fix i E Nl and let k := ri. Then there exists B E JZi such that, for

( (B(N) , A i + l , ' " , Al)) = Fi(k, N).

Proof." Let b: N --~ Z be defined by

b(N) := min{u E N iupi(A~) e N Z } .

By Lemma 6.4, b is h.l.p. Now define B: N ~ H(Z) by B ( N ) := Abi (N).

LEMMA 8.4: Let i E Nl and let k E S~. Then there exists B E Fi such that, for

a l l N E N,

( (B(N) , A i + l , ' " , At)> = Fi(k, N) .

Proof: The proof is by induction on k. By Lemma 8.3, we may assume that

ri < k < si.

By the induction hypothesis, there exists C E ~-i such that, for all N E N, we

have

<<C(N), A i + l , . . . , A,)> = Fi(k - 1, g ) .

The map ~rk o C: N --* Z is 0-v.p.p. By Propositions 5.9 and 5.10, choose an

h.l.p, a: N --~ Z such that, for all N E N,

:rk(C(N)) =- a ( N ) m o d N.

Define b: N --* Z by b(N) := min{u e N Iua(n) E NZ}.

periodic. Define B E 9vl by B ( N ) := C(N) b(N). 1

By Lemma 6.3, b is

LEMMA 8.5: Let i , j E Nt and assume that i <_ j. Let k E Sj. Then there exist

Bi E Jri , . . . , Bj E Jrj such that, for a11 N E N, we have

( ( B i ( g ) , . . . , B j (N) , A j + I , . . . , Az)) = Fi(k, g ) .

Proo~ The proof is by induction on k - ri. By Lemma 8.4, we may assume that

i < j .

CASE 1: k ='rj.

By induction, choose Ci+l E ~ i + 1 , . . . , Cj E J:j such that, for all N E N, we

have

( (C~+l (g ) , . . . , C j (N) , A j + I , . . . , Az)) = Fi+l(k, Y ) .


Since k = rj, it follows that k - 1 E Sj-1. By induction, choose Di E .Ti, . . . , Dj-1

E ~' j -1 such that , for all N C N, we have

( (Di(N) , . . . , Dj-I (N) , A j , . . . , Alll = Fi(k - 1, N).

Let t := j - i. The maps 7rk o Di , . . . , ~rk o Dj_ 1 are 0-v.p.p. By Proposi t ions 5.9

and 5.10, choose h.l.p, functions a, b l , . . . , bt-l: N --~ Z such that , for all N E N,

a(N) - 7rk(D~(g))mod N,

bx(N) - 7rk(Oi+l(N))mod N,

. , .

bt-l(N) =- rk (Dj - l (N) )mod N.

Define bt, c: N ~ Z by bt(N) := N and c(N):= 7rk(Aj). For all N E N, let

a'(N) := min{u e N Iua(g ) e b l ( g ) z + . . . + bt(N)Z + c(N)Z}.

Then, for all u, N E N,

(*) ua(N) �9 b l ( N ) Z + . . - + bt(N)Z + c ( N ) Z ~ u �9 a ' ( g ) z .

By Lemma 6.6, we find tha t a ' : N --+ Z is periodic and we may choose periodic

functions b~ , . . . , b~: N --* Z and an h.l.p, function d: N --~ Z such that , for all

N �9 N, we have

a(N)a'(N) = bl(N)b~l(N) + . . . + bt(N)b't(N ) + c(N)c'(N).

For all N �9 N, define

B~(N) := D~(N)a'(N)Di+x(N) b'~(N)... D. . (N ~b~-l(N)~c'(N)

Bi+I(N) := Ci+l(N) , . . . , Bj(N) := Cj(N);

then B d N ) , . . . , Bj(N) �9 Fi(k, N). Fix g �9 N and E �9 Fi(k, g) . To finish

CASE 1, we must show that

E �9 ( (B~(N) , . . . , Bj(N), Aj+I , . . . , Ak>>.

Since E �9 F{(k, N) C_ Fi(k - 1, N), there exist nl , . . . , nz �9 Z such tha t

E = D,(N) n ' . . . Dj-1 (g) nj-1A~ ~. . . A'~ ~.


For all N E N, let RdN: Z ~ Z / N Z denote reduction modulo N; then

(RdN o 7rk)l{h �9 H I ~r l (h) , . . . ,~rk- l (h) �9 NZ}

is a homomorphism. Furthermore, 7rk(E) �9 N Z and bt(N) = N, VN �9 N; it

follows from (*) that ni �9 a ' (N)Z . Choose m �9 Z such that ni = a'(N)m. By

Lemma 7.1, p~ is a homomorphism, which implies that pi(Bi(N)-mE) = 0, or

Bi (N) -mE �9 F~+,(k, N) = ((Bi+l(N), . . . , Bj(N), Aj+I , . . . , Al)).

It follows that E �9 ((B~(N),. . . , Bj(N), A j + I , . . . , Al)), as desired to finish CASE

1.

CASE 2: rj < k <_ sj.

By induction, choose C~+I �9 F~+I,..., Cj �9 ~j such that, for all N �9 N, we

have

( (CI+I (N) , . . . , C j (N) , A j + I , . . . , Al)) = F i+ l (k , N) .

Since rj < k <_ sj, it follows that k - 1 �9 Sj. By induction, choose Di �9

.T~,..., Dj �9 .Tj such that, for all N �9 N, we have

((D~(N),. . . , Dj(N), Aj+I , . . . , Az)) = Fi(k - 1, N).

Let t := j - i + 1. By Propositions 5.9 and 5.10, choose h.l.p, functions

a, b l , . . . ,b t - l : N --~ Z such that, for all N �9 N, we have

a(N) - rk(Di(N))rood N,

hi(N) - 7rk(Di+l(N))mod N,

bt_,(N) -- ~rk(Dj(N))rood N.

Let bt: N --* Z be defined by bt(N) :-= N.

For all N �9 N, let

a'(N) := min{u �9 N]ua(N) �9 b l (N)Z + . . . + bt(N)Z}.

By Lemma 6.5, we find that al: N --* Z is periodic and we may choose periodic

functions b~ , . . . , b~: N --* Z such that, for all N �9 N, we have

a(N)a'(N) = bl(N)b~(N) + . . . + bt(N)b't(N ).


For all N �9 N , define

Bi(N) := Di(N)a'(N)Di+I(N) b'~(N)... Dj(N) b't-~(N),

Bi+l(g) := C i + l ( g ) , . . . , Bj(N):= C j ( N ) ;

then B~(N), . . . , Bj(N) �9 Fi(k, g) .

The conclusion of CASE 2 is very s imi lar to the conclusion of C A S E 1; we omi t

it. I

Recal l f rom L e m m a 7.1 t ha t pi := 7r~, ]Fi(Z) .

COROLLARY 8.6: There exist B1 �9 .T'I,..., Bt C .T't such that: for all i �9 Nt, for

all N �9 N, we have

Fi(m, N) = ((Bi(N), . . . , Bt(N))).

Proof." By the i -- 1, j = l, k = m case of L e m m a 8.5, choose B1 �9 J r l , . . . , Bz �9

5~t such tha t , for all N �9 N , we have

Fl(m, N) = ( ( B I ( N ) . . . . , Bt(N))).

Fix i �9 Nz, N �9 N and E �9 Fi(m, N); we wish to show tha t

E �9 ( (Bi(N) , . . . , B,(N))).

Since E �9 Fl(m, N), choose n l , . . . , nt �9 Z such t ha t E = B I ( N ) n~ . . . Bt(N) TM.

If i = 1, then we are done. So assume i > 1. Then p l ( E ) -- 0 and so n l = 0.

So if i = 2, then we are done. So assume i > 2. Then p2(E) -- 0, so n2 = 0. If

i -- 3, then we are done.

Cont inuing this way, the proof will eventua l ly be complete , for any value of i.

8

For each N �9 N , let RdN: H ( Z ) ~ H(Z/NZ) denote reduct ion m o d N and

let H(N) := k e r ( R d g ) .

For any subset S of a group, we denote the subgroup genera ted by S as (S).

For all v �9 N , for all M 1 , . . . , M v �9 SLn(Z) , let ( M 1 , . . . , M v ) denote the

subgroup of SLn(Z) genera ted by M 1 , . . . , M . .

COROLLARY 8.7: There exists a collection ofh.l.p, functions

aj . i" N - * Z ( i , j E N l , i _< j )


such that, for ali N E N, we have

H(N) := <A~ :(N)...AT~(N)]i E Nt>,

and such that, for all i E Nt, a~ is positive.

Proo~ Choose B1 E ~-1 . . . . , Bt E ~-l as in Corollary 8.6.

Fix i E Nl. Define b~: N --* Z by

b~(N) := p,(B,(N))/pi(A,);

b~ is h.l.p. Since Bi E U~ and since p~(A~) > O, it follows that b~ is positive. For

each j = i + 1 , . . . , l, define b~: N --* Z inductively by the recursive formula

: - -p j~Aj_ -b~ I(N) ...ATb~(N)Bi(N)) / p j (A j ) ; b}(N)

b} is 0-v.p.p.

Then, for all i E Nt and N E N,

b~(N) A~(N). BI(N) = A i "'"

Fix i E Nl. Define a~ := b~. Using Propositions 5.9 and 5.10, choose h.l.p, func-

tions a~+x,.. . , a~: N --~ Z such that, for all N E N, we have

t a}(N) -- b}(g)mod N(n!) 2. Vj E Zi+l,

It is an exercise to use the expansions of the exponential and logarithm maps

on unipotent matrices to show: If M is an upper triangular, unipotent n • n

matrix, if s, t E Z, if N E N and if s --- t rood N(n!) 2, then M s =- M t rood N.

Applying this, we find: for all i E Nz, for all j E Z~, for all N E N, we have

A~ }(N) _= A~ "~(N) mod N.

.a~(N) Let Ci(N) := Ai . . . At ~(g). Then, for all i E Nl, for all N E N,

C~(N) = A a~(N) . . . A7 ;(N) =- A~ :(N) ' ' " A~ ;(N) = Bi (Y)mod N,

so C~(N) E Fi(m, N).


CLAIM: For al1 i C Nt, for all N E N,

Fi(m, N) = ( c i ( g ) , . . . , C,(N)).

The proof of the CLAIM is by (downward) induction on i = l , . . . , 1. When

i = l, the result follows from Corollary 8.6, because Cl(N) = At [(g) = Bt(N).

Now fix i E Nt -1 , N E N and E E Fi(m, N); we wish to show tha t

E E <c~(g), . . . , c , ( g ) ) ,

under the induct ion assumption

F~+l(m, N) = (C~+I(N),.. . , cz (g )> .

By Corollary 8.6, we may choose n~ , . . . , nz E Z such tha t

E = B~(N) n ' . . . Bz(N) TM.

Since a~(N) = b~(N), it follows from Lemma 7.1 tha t

so C i ( Y ) - ~ ' Z �9 Fi+l(m, N) = <Ci+l(Y),. . . , c t ( g ) ) , which proves the CLAIM.

But then the special case i = 1 of the CLAIM reads: For all N �9 N,

( c ~ ( g ) , . . . , c t ( g ) ) = F~(m, N) = H(N).

By definition of C 1 , . . . , Ct, this is exactly what we wish to prove. 1

9. R a t i o n a l p lanes and p o l y n o m i a l per iod i c i ty

Let H and 1 be as in w For the remainder of this section, fix a positive integer

d.

Let T O denote the trivial group.

Fix c E N. Let T c denote the c-torus, thought of as a multiplicative, compact ,

Abelian group. Let tor T c denote the subgroup of all elements in T ~ of finite

order. A subset 7r C_ T ~ is a rat iona l p lane if there exists an element ~ E tor T ~

such tha t r is a closed, connected subgroup of T ~.

Vol. 88, 1994 BETTI NUMBERS OF CONGR U EN C E GROUPS 61

LEMMA 9.1: Let c E N and let b: N ~ Z be O-v.p.p. Let r C T ~ be a rational

plane and let w E t o r t ~. For ali N E N, let f ( N ) be the cardinality of the set

{X E 7[IX b(N) = w}.

Then f: N ---* Z is O-v.p.p.

Proo~ Choose ( E tor T c such tha t S := Qr is a closed, connected subgroup

of T c. Let e := d i m S . Let u denote the order of (. By Propos i t ion 1.9,

there exists a periodic function b~: N ~ Z such tha t , for all N E N, we have

b(N) =_ b'(N) m o d u.

Fix N E N. Then

is 1-to-l , while

is b(N)~-to-1, while

X ~ (X: ~r -+ S

~ r S ~ lr b(N)

is 1-to-1. We conclude tha t the composi te m a p

)~ ~ )/b(~): 7r ~ Ir b(g)

is b(N)~-to-1.

For all N E N , we have 7r b(N) = (-b(N)s; define

a(N) := { O, if w q~ (-b(N)s 1, if w E r

For all N E N, we have (-b(N) = ~-b'(N). Since b' is periodic, the funct ion

a: N --* Z is periodic. Now, for all N E N , we have f ( N ) = a(N)b(N)~; it

follows tha t st: N --* Z is 0-v.p.p. , as asserted. I

Definition 9.2:

define

For all c E N , X = ()~1,- . . , Xc) ~ TC, o~ -- ( o Q , . . . , o~c) E Z c, we

(X,o~) : : X~I...Xc~o E T 1.


Definition 9.3: Let X : ( X 1 , " ' ' , Xd) E W d. Let i , j E Nd and assume i < j . We

define )/~j := ()~i, �9 �9 )Cj) E T j -~+l . We will abbreviate )~i~ as )/~, for simplicity.

Let T~ denote the set of all rat ional planes in T d.

For all c, k E N satisfying c _~ k, let S(c, k) denote the set of all (/~1,. . . , ~k) E

(ZC) k such tha t Z c = ~1 z -~- . . �9 ~- ~ k Z .

Definition 9.4: Let b: N --* Z be 0-v.p.p. Let c E Nd, k E Nl satisfy c _< k.

Let ~r E 7~, ~ = ( ~ l , . . . , ~ k ) E S(c,k) . Let s = ( S l , . . . , s k ) E t o r T k, w =

(we+ l , . . . ,Wd) E t o r T d-c. For all N E N, let E (N) denote the set of all )/ E 7r

such tha t

d . b(N) Vi E Nk, (~lc, b(N)~i) = si and Vj E Zc+ ~, Xj = wj.

We define Fb~8~: N --* Z by letting Fb~8~(N) be the cardinali ty of E(N) .

Similarly, for all N E N, we let $'(N) denote the set of all )~ E 7r such tha t

v j �9 Zl _- w j

We define ' �9 F g ~ . N --~ Z by letting F ~ ( N ) be the cardinali ty of C ' (N) .

We define T O to be the trivial group (1). When c = d and w = 1, we define

Fb,~8~ as in Defintion 9.4, but with the convention tha t E (N) denotes the set of

all ) / � 9 7r such tha t

Vi �9 Nk, (Xld, b(N)Zi) = si.

De/~nition 9.5: Let b: N ~ Z be 0-v.p.p. Let c �9 Nd, k �9 Nt. If c _< k, then we

define 9rb(c, k) to be the Z-span of the set of functions

( F b , z ~ I 7r �9 n , ~ �9 S(c ,k) , s �9 t o r T k, w �9 t o rTd-~} .

If c > k, then we define J:b(C, k) to be the set consisting of 0 alone, where

0: N ~ Z is the function which is identically zero. Finally, we define ~'g to be

the Z-span of

( / 7 ~ IIr �9 n , w �9 t o r T d - C ) .

LEMMA 9.6: Let b: N --* Z be O-v.p.p. Let c �9 Z d, k �9 Zl2 satisfy c ~ k. Then

.~b(c, k) c_C_ ~'b(C -- 1, k - 1) U .~b(C, k - 1).

Proof." F i x c � 9 d , k � 9 1 4 9 1 4 9 k,

w �9 t o r T d-~. Let g := Fb~z~; we wish to show tha t g �9 : F b ( C - l , k - 1) U

J :b ( c , k - 1) .

Vol. 88, 1994 B E T T I NUMBERS OF C O N G R U E N C E GROUPS 63

If/3k = 0 and sk ~ 1, then g: N --* Z is identically zero and we are done.

Let ~ := (/31, . . . , /3k-1) and ~ := ( S l , . . . , S k - 1 ) . If/3k = 0 and sk = 1, then we

have g = F b ~ ~ E 9cb(C, k - 1) and, again, we are done.

We may therefore assume tha t /3k ~ 0. We will prove in this case tha t g E

.Tb(C- 1, k - 1).

Let c := (0, 0 , . . . , 0, 0, 1) E Z ~. Choose an au tomorphism ~: Z r --, Z ~ such tha t

/3k E y(e)Z. Let I : Z d-~ --* Z d-c denote the identity map. Let 0 := ~ • I : Z d -~ Z d"

Then ~ induces an au tomorphism ~ := ~k: (Zr __~ (Z~)k, and 0 induces

an adjoint au tomorph ism ~: T d -~ T d defined by (~(X), O(a)) = (X, a) , for all

X E T d and all a E Z d.

Replacing ~r by (~)-1(7r) a n d / 3 by (~)-1(/3), we may assume that /3k E oZ.

Choose h i , . . . , nk E Z and "Yl,.. . , 7k-1 E Z ~-1 such tha t

/31 = (~fl, n l ) , . . . , /3k-1 = (7k-1, nk-1) , /3k = (0, nk).

Let 5 := c - 1. Then, for all N E N, g ( N ) is the number of X E ~r satisfying

d . b (N) Vi E Nk-1 , (Xih, b(N)~i)xbc (N)n' = sl, xbc (N)nk = sk, Vj E Z~+l, Xj = wj .

Let V := {v E t o r T l l vn~ = sk}. For a l l v E V, for a l l N E N, let h , ( N )

denote the number of X E n satisfying

d b(N) Vi E Nk-1 , (Xle, b(N)"/i> = s i v -n ' , X b(N) = v, Vj E Zc+l, Xj : Wj.

Then, for all v E V, hv E .Tb(C- 1, k - 1) and it follows from inclusion-

exclusion tha t g is a Z-linear combinat ion of elements of {hv I v E V}. Thus g E

~ b ( C - 1, k - 1), as desired. I

We omit the proof of the next lemma since its proof is similar to, but easier

than tha t of Lemma 9.6.

LEMMA 9.7: Let b: N --, Z be O-v.p.p. Le t k E Nd. Then ~-b(1, k) C_ YrS.

COROLLARY 9.8: Let b: N --* Z be O-v.p.p. Let c E N d , k E Nl . Then ~b(C, k) C_

Proof: The proof is by induction on k. If c > k, then $-b(c, k) consists of the

zero function alone, and we are done. If c = 1, then we are done by Lemma 9.7.


We therefore assume 2 < c < k. The induction hypothesis and Lemma 9.6 now

complete the proof. I

LEMMA 9.9: Let b: N --* Z be O-v.p.p. Let 7r E R, ( a l , . . . , a t ) E S(d,l) ,

( r l , . . . , rt) E torT t. For all N E N, let f ( N ) denote the cardinality of

{x �9 I vi �9 Nl, (x ,b(N)ai) = r,}.

Then f: N ~ Z is O-v.p.p.

Proof: Let a := ( a l , . . . , a t ) , r := ( r l , . . . , r ~ ) and v := 1 �9 T O . Then

f = Fb~rarv �9 fib(d, l),

SO, by Corollary 9.8, f �9 ~-~.

From the definition of ~'~ and from Lemma 9.1, it follows that ~-~ is the Z-span

of a set of 0-v.p.p. functions. Consequently, every element of ~'~ is 0-v.p.p.; in

particular, f is. I

COROLLARY 9.10: Let 7r �9 TZ. Let b: N ~ Z be O-v.p.p. and let ( a l , . . . , a t ) �9

S(d, l). Let 7" denote the collection of all/]nite subsets of torT t. Let T: N ~ 7"

be a periodic function. For all i �9 Nt, for all N �9 N, let r N) := (X, b(N)a~).

For a11 N �9 N, let f ( N ) denote the cardinality of the set

{X �9 7r I (r N ) , . . . , r N)) �9 T(N)} .

Then f : N --* Z is O-v.p.p.

Proof'. If T: N ~ 7" is constant, then the result follows from Lemma 9.9 and

the fact that a sum of 0-v.p.p functions is again 0-v.p.p.

Choose m �9 N such that, for all i �9 N,,~, T is constant on (mZ + i) fq N.

Then, for all i �9 Nm, the constant case demonstrates that f agrees with a 0-

v.p.p, function on (mZ + i) O N.

I t follows that f is 0-v.p.p. I


10. A t e c h n i c a l r e s u l t

Let H be as in w For each i E Nl, let Ai be as in w

For all N E N, let H(N) denote the kernel of the map H(Z) --* H ( Z / N Z )

defined by reduction mod N.

THEOREM 10.1: Let F denote the Abelianization of H(Z) . Let F denote the

dual group to F and let ~o denote its connected component of the identity; ~o

is a (compact, connected) torus group. Fix an element )Co C F. Fix a rational

plane 7r C_ ~o. For all N E N, let F(N) denote the image in F of H(N). For

all N e N, let f ( g ) denote the cardinality of {X e 7r I (xox ) l r (N) = 1}. Then

f: N --* Z is p.p.

Prool~ We will denote the group operation on F as addition. Let F t~ and F tf

denote the torsion and torsion-free parts of F, respectively.

Let pr: F --* F tf denote the natural projection and define )C1 := Xo o pr. Then

(XoX~-I)I Ftf = 1 and X1 E ~o. Replacing Xo by XoX~ -1 and ~r by xl~r, we may

assume that XoIF tf = 1.

Let d := rank(Ftf). Choose an isomorphism between F tf and zd; this gives

rise to an isomorphism between ~o and the multiplicative group T d.

Let a}: N --* Z (i , j e Nz, i _< j ) be as in Corollary 8.7. Let a l , . . . , az denote

the images of A1 , . . . , At in F. Let a ~ ~ ~ denote the torsion parts of

a l , , az. Let a tf, . , a t f denote the torsion-free parts of a l , �9 �9 al. Note that

( a t f , . . . , a tf) E S(d, l), after the identification of r t f with Z d.

For all i E Nl, for all N E N, let

fli(N) := a~( N)a~ + . . . + a~(N)al.

Then, for all N E N, F(N) = f l l (N)Z + . - . + ~z(N)Z, so f ( N ) is the cardinality

of

S(N) := {X e ~r[ ()CoX, t31(N)) . . . . . (XoX, ~t(N)} = 1}.

For all i E Nl, for all N E N, let

bi(N) := gcd(a~(N) , . . . , a~(N)).

For all N E N, let b(N) := g c d ( b l ( N ) , . . . , bl(N)). Then b: N --* Z is positive

i a}/b; then and h.l.p., by Corollary 6.2. For all i C Nl, for all j E Z~, let qj :=

i. N ~ Z is periodic. qj.


For all i �9 Nl, define r F > N --* T ' by r = (x, b( N)a~). Recall

that ( a t f , . . . , a[ f) �9 S(d, l). As in Corollary 9.10, for all i �9 N~, define ctf : ~ )<

N ~ W 1 by c t f ( x , N ) = (x ,b(g)a~f) �9 For a l l / �9 Nt, for all N �9 N, let

c t o r , N" b(g)a~~ ()~, ) := (X,

Fix i �9 Nt. For all X �9 ~0, for all N �9 N, we have

r (Xxo, N)r (Xxo, N) N ) r r N) = tor tf _,tor~ = 'q~i (XO,

Let ki be the order of _tor By Proposition 9.5, choose a periodic function

b~: N --~ Z such that b~(N) = b(g) mod ki. Then, for all N �9 N, c t~ N) = b~/N~ator\ - t o r , N): N T 1 Xo, i~ J i /, so the function N H ~i (X0, ~ is periodic.

Let T denote the set of all finite subsets of tor T t. For all N �9 N, for all

i �9 Nt, for all ~ = (~1, . . . , ffl) �9 T t, define

A~(~, N) := ~q~(N) ~7~(N).

For all N �9 N , let

To(N) := {~ �9 I t I AI((, N) . . . . . A,((, N) = 1},

T ( N ) := (r176 Y ) - l , . . . ~bt~ g ) - l ) T o ( g ) .

The functions

For all N �9

S(N) =

But, for all

Corollary 9.10.

To, T: N ~ T are periodic.

N, we have

{X C 7r I ViE Nl, r N) q'~(N)'''r q~(N) = 1}

{X E 71" I (r N ) , . . . , Ct(XoX, N)) E To(N)}

{X e 7r I (e l f (x , N) . . . . . ~tf(x, N)) �9 T(N)}

N �9 N, f ( N ) is the cardinality of S(N); the result follows from |

11. R e d u c t i o n to t h e u p p e r t r i a n g u l a r case

LEMMA 11.1: Let n C N. Let v l , . . . , v n be an ordered Q-basis for Qn. Then

there is an ordered Z-base Wl , . . . , w~ for Z ~ such that the full flag is unchanged,

i.e., such that

Q v l = Q W l , Q v l + Q v 2 = Q w 1 4 - Q w 2 , . . . ,

Qvl + - . - + Qvn = Qwl + . . . + Qw,~.


Proof: We construct the wis inductively. Let wl be a generator of the infinite

cyclic group Z n Cl (Qvl) .

Now assume that Wl , . . . ,Win E Z" have been chosen so that

ZVl + ' ' " q- ZVm = Z n N (QVl + ' - ' - F Qvm)

and so that

QVl = QWl, Qvl + Qv2 = Qwl + Qw2, . . . ,

Q V l + . . . + Qvm = Qwl + - . - + Q w m .

The image C of Z" N (QVl + . . . + QVm+l) in the one-dimensional quotient vector

space

(Qvl + . . . + QVm+l) / (qv l + . . . + qvm)

is nonzero and discrete, hence infinite cyclic. Let c be a generator of C and let

Wm+l be a preimage in Z n N (QVl + " " + Qvm+l) of c. |

COROLLARY 11.2: Let n be a positive integer. Let H be a unipotent algebraic

subgroup of SLy. Then there exists an element w E SLn(Z) such that w - l H w

is contained in the upper triangular unipotent matrices in SLy.

Proof: Let V denote the (algebraic) vector space A s• 1 of n x i column matrices.

Let V1 ~ denote the Q-subspace of V consisting of those vectors fixed by every

element of H under matrix multiplication. By [Hum, Corollary 17.5, p. 113],

171 ~ {0}. Since it is a linear subspace defined over Q, we may choose a Q-

point vl E Vl\{0}. Let V1 denote the linear space spanned by vl. Then H

acts by special linear transformations on V/V1. Furthermore, the image of H in

SL(V/V1) is again unipotent. By using induction on dim V, we obtain Q-vectors

v l , . . . , vn with linear spans V1, . . . , V~ such that every element of H fixes the full

flag

o c_ Yl C V l | C " c_ v l | = y .

Let v denote the matr ix with columns v n , . . . , Vl. It follows that v - l i l y consists

of upper triangular unipotent matrices.

Then Lemma 11.1 (with Q~ replaced by the isomorphic vector space QnXl)

yields column vectors Wl , . . . , wn E Z nxl. Replacing Wl by -Wl if necessary, we

may assume that the n x n matr ix w with columns w ~ , . . . , wl satisfies det w =


1. Then w - l i l T = v - l i l y , so w - l i l T is contained in the upper triangular

unipotent matrices in SLn. |

The following technical result has been the main goal of w167

THEOREM 11.3: Let n be a positive integer and let H be a unipotent Q-subgroup

of SL,~. Let r denote the Abelianization of H(Z) . Let F denote the dual group

hom(F, T 1) of r , where T 1 C C denotes the complex numbers of modulus one.

Fix an element Xo E F. Let ~o denote the connected component of the identity

in F. Let r C_ ~o be a translate of a closed connected subgroup of ~o by: an

element in ~o of finite order. For ali integers N > O, let Rdg: H(Z) ~ H ( Z / N Z )

be the map defined by reduction rood N and let H ( N ) denote the kernel of RdN;

H ( N ) is the N t h c o n g r u e n c e s u b g r o u p of H. For each N > O, let F(N)

denote the image of H ( N ) in the Abelianization F of H(Z) . For each N > O, ]et

f ( N ) denote the number of elements X E lr such that (XoX)lr(N) is identically

1. Then the function f: N -* Z is p o l y n o m i a l pe r iod i c . That is, there exist

(1) a positive integer m; and

(2) a finite set of polynomial functions f x , . . . , fro: R --+ R

such that: i f q and r are integerssatisfyingq >_ 0 and I < r < m, then f ( q m + r ) =

fr(q).

Proof'. This follows immediately from Theorem 10.1 combined with Corollary

11.2. |

[Ad]

[B-W]

[C-R]

[F-T]

[F-l]

References

S. Adams, Representation varieties of arithmetic groups and polynomial pe-

riodicity of Betti numbers, Israel Journal of Mathematics, this issue, pp.

73-124.

A. Borel and N. Wallach, Continuous cohomology, discrete groups and rep-

resentations ofreductive groups, Annals of Math. Studies 94, Princeton Uni-

versity Press, 1976.

C. Curtis and I. Reiner, Methods of Representation Theory, Wiley, New

York, 1981.

R.H. Fox and A. Torres, Dual presentations of the group of a knot, Annals

of Mathematics 89 (1954), 211-218.

R. H. Fox, A quick trip through knot theory, in Topology of 3-Manifolds

(M. K. Fort, Jr., ed.), Prentice-Hall, Englewood Cliffs, NJ, 1961.

Vol. 88, 1994

[F-2]

[G-G-P l

IN]

[Ha]

[Ha2]

[na3]

[nh]

[I]

[Jac] [g]

ILl]

[L2]

[Lt]

[Lr]

[Lr2]

[L-M]

[M-M]

[P-S]

[R-S]

BETTI NUMBERS OF CONGRUENCE GROUPS 69

R. H. Fox, Free differential calculus I, II, III, Annals of Mathematics 364

(1956), 407-447.

I. Gelfand, M. Graev and I. Piatetsky-Shapiro, Representation Theory and

Automorphic Functions, W.B. Saunders, London, 1969.

L. Georite, Die Bettischen Zahlen der Zyklische Voerlangerung der Knote-

nauschenraume, American Journal of Mathematics 56 (1934), 194-198.

E. Hironaka, Ph.D. Thesis, Brown, 1989.

E. Hironaka, Polynomial periodicity for Betti numbers of covering surfaces,

Inventiones Mathematicae 108 (1992), 289-321.

E. Hironaka, Intersection theory on branched covering surfaces and polyno-

mial periodicity, preprint.

F. Hirzebruch, Arrangements of lines and algebraic surfaces, in Arithmetic

Geometry, Vol. II (M. Artin and J. Task, eds.), Birkh/iuser, Boston, 1983.

M. N. Ishida, The irregularities of Hirzebruch's surfaces of general type with

C12 -- 3 C2, Mathematische Annalen 262 (1983), 407-420.

N. Jacobson, Basic Algebra II, W. H. Freeman and Co., San Francisco, 1980.

M. Kneser, Strong approximation, in Algebraic Groups, Proceedings of

Symposia in Pure Mathematics IX (1966), 187-196.

S. Lang, Introduction to Algebraic and Abelian Functions, Springer-Verlag,

Berlin, 1982.

S. Lang, Fundamentals of Diophantine geometry, Springer-Verlag, Berlin,

1983.

M. Laurant, Equations diophantine exponentielles, Inventiones Mathemati- cae 78 (1984), 299-327.

A. Libgober, Betti numbers of Abelian covers, Preprint, 1989.

A. Libgober, Alexander polynomials of algebraic curves, Duke Mathematical

Journal 49 (1982), 833-851.

A. Lubotzky and A. Magid, Varieties of representations of finitely generated

groups, Memoirs of the American Mathematical Society58 (1985).

J. Mayberry and K. Murasugi, Torsion groups of Abelian coverings of links,

XYansactions of the American Mathematical Society 271 (1982), 143-173.

R. Phillips and P. Sarnak, The spectrum of fermat groups, Geometry Func-

tional Annalysis 1 (1991), 80-146.

D. Ray and I. Singer, R-torsion and the Laplacian, Advances in Mathematics

7 (1975), 145-210.

7O

[R]

[Ru]

[Sa] [s]

[Su]

[z]

P. SARNAK ET AL. Isr. J. Math.

Z. Rudnick, Representation varieties of solvable groups, Journal of Pure and

Applied Algebra 45 (1987), 261-272.

W. Ruppert, Solving algebraic equations in roots of unity, Journal ffir die

Reine und Angewandte Mathematik 435 (1992), 119-156.

P. Sarnak, Betti numbers of congruence groups, preprint.

B. Schoenberg, Elliptic Modular Fanctions, Springer-Verlag, New York,

1974.

D. Sumners, On the homology of cyclic coverings of higher dimensional links, Proceedings of the American Mathematical Society 46 (1974), 143-149.

O. Zariski, On the topology of algebraic singularities, American Journal of

Mathematics 54 (1932), 453-465.

Appendix: On Representations of Compact p-adic Groups

Let G be a Chevally group, G(R) the corresponding group over the ring of integers

R of a p-adic field of characteristic zero. In this note we give an a priori proof of

the finiteness of representations of G(R) in a given dimension; here, as throughout

the rest of this note, all representations are assumed to be continuous.

We begin by noting that it suffices to prove this theorem for some normal open

subgroup F <~ G(R)--since if p is any irreducible representation of G(R), then p

injects into the induced representation:

p r Indr~(n) Rest ~(n) p

and because G(R)/F is a finite group, there are only finitely many possibilities

for p given its restriction to F.

From now on, for simplicity of exposition, I take G = SL(N), R = Zp, the

p-adic integers, and p r 2, 3. Let

Fe = F(p e) = {7 e SL(N, Zp) I 7 - I mod p%[(g, Zp)}

be the principal congruence subgroup of level pe; here s[ (N, Zp) is the Lie algebra

of SL(N, Zp). We take F = F1.

If p is a continuous finite dimensional representation of F, the l e v e l / p of p is

the least integer g _> 1 such that p is trivial on Fe. In this case, p factors through

the finite group Fe = F/Fe.

The finiteness result will follow from the following estimate:

Vol. 88, 1994

PROPOSITION A.I :

and level 2. Then

(A.1)

BETTI NUMBERS OF CONGRUENCE GROUPS 71

Let p be an irreducible representation of F of dimension n

<< J(n) 1/l~ -- f (n )

where J(n) is the index of a normal, Abelian subgroup of p(r) = p(Gt); J(n)

depends only on n, not on p.

Indeed, assuming (A.1), any p of dimension n will factor through one of a finite

number of finite groups, so the number of such representations is finite.

To prove (A.1), we need a couple of lemmas. First, recall that since Gt is a

p-group, it is in particular solvable (even nilpotent).

LEMMA A.I : G~ has solvable length given by

(g.2) length (G~) = [log2(s 1)] + 1, ~ _> 2,

i.e., i f2 k-1 < ~ _< 2 k then length (G~) = k.

Proofi This follows from computing the commutator subgroups of F:

(rk, rk) = r2k.

The inclusion (Fk, Fk) C_ r2k is obvious, and equality follows by applying the

Campbell-Baker-Hausdorff formula. It is here that one has to use the exponential

exp: sI (N, PZv) -* F.

Recall that the exponential series:

oo x n

exp(x) = ~ n = 0

is p-adically convergent only for [X[p < 1. |

LEMMA A.2: I f g is the level of p, and n its dimension, we have

(A.3) length (Gt) _< 1 + logp J .

Proof: First note that since p is trivial on Ft, but not on F~-I, we have

length p(F) = length Gt.


However, if A <1 p(F) is a normal abelian subgroup of index _< J, then:

lengthp(F) < lengthA + leng thp( r ) /A

= 1 + lengthp(F) /A

< 1 + logp J . |

Putt ing together (A.2) and (A.3), we find:

log 2 p << logp J

which proves (A.1), except that we have to show that J can be taken independent

of p. This is precisely the content of Jordan's theorem [CR], which asserts that

any finite subgroup of GL(n, C) contains the normal Abelian subgroup of index

at most

(A.4) J(n) = ( v / ~ + 1) 2n2 - ( v f ~ - 1) 2n' .

This proves (A.1). |

Remark: The estimate (A.4) is very far from the truth; Howe's Theory [HI

provides us with a normal Abelian subgroup of index (exactly!) n 2.

R e f e r e n c e s

[CR] C. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative

Algebras, Wiley-Interscience, New York, 1962.

[HI R. Howe, Kirilov theory for compact p-adic groups, Pacific Journal of

Mathematics 73 (1977), 365-381.

Documents

Betti numbers of congruence groups