Commensurabilities among Lattices in PU (1,n). (AM-132), Volume 132 9781400882519

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Annals of Mathematics Studies

Number 132

Commensurabilities among Lattices in PU(l,n)

by

Pierre Deligne and G. Daniel M ostow

P R IN C E T O N U N IV ER SITY PRESS

P R IN C E T O N , N E W JERSEY 1993

Copyright © 1993 by Princeton University Press ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein

Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources

Printed in the United States of America

Library of Congress Cataloging-in-Publication Data Deligne, Pierre. Commensurabilities among lattices in PU(\,n) / by Pierre Deligne and G. Daniel Mostow. p. cm. — (Annals of mathematics studies; no. 132) Includes bibliographical references. ISBN 0-691-03385-4 — ISBN 0-691-00096-4 (pbk.) 1. Functions, Hypergeometric. 2. Monodromy groups. 3. Lattice theory. I. Mostow, George D. II. Title. III. Series QA353.H9D45 1993 515'.25—dc20 93-5528

The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed

CONTENTS

§1. In tro d u ctio n

1

§2. P ica rd G roup an d C o h om ology

10

§3. C o m p u ta tio n s for Q an d Q+

17

§4. L a u ricella’s H y p e rg eo m etr ic F u n ction s

27

§5. G elfa n d ’s D e sc rip tio n o f L au ricella’s H y p e r g e ­ o m e tric F u n ction s

35

§6. S trict E x p o n en ts

43

§7. C h a ra cteriza tio n o f H y p erg eo m etric-lik e L ocal S y stem s

55

§8. P relim in a ries on M on od rom y G rou ps

71

§9. B a ck grou n d H eu ristics

80

§10. S o m e C o m m en su ra b ility T h eo rem s

84

§11. A n o th e r Iso g en y

102

§12. C o m m en su ra b ility an d D isc r e te n e ss

119

§13. A n E x a m p le

124

§14. O rbifold

135

§15. E llip tic an d E u clid ean /i’s, R e v isite d

142

§16. L iv n e’s C o n stru ctio n o f L a ttices in P U ( 1,2)

161

§17. L ine A rra n g em en ts o f C om p lex R e flectio n G roups: Q u estio n s

169

B ib lio g rap h y

182

ACKNOWLEDGMENTS

T he authors are grateful to Dr. Rolf-Peter Holzapfel whose careful read­ ing of a prelim inary version of this monograph helped us elim inate a num ber of m isprints. The authors wish to acknowledge our debt to Ms. Donna Belli whose expertise and patience transform ed our m anuscript and diagram s into camera-ready form.

Commensurabilities among Lattices in PU(l,n)

51. INTRODUCTION

The aim of this m onograph is to investigate lattices in P C /(l,n ) groups. Fix n > 1 and let Q be the moduli space of (n -f 3)-uples of distinct points on a projective line: the quotient by P G L { 2) of ( p 1)n+3 minus the diagonals X{ = Xj.

R ather than dividing by P G L ( 2), one may fix

xn+i = 0,arn+2 = l ,^ n +3 = oo. This identifies Q with the space M of n-uples ( x i , . . . , x n ) of distinct points of P 1, with xt- ^ 0,1, oo. Let fj, be a (n + 3)-uple of complex numbers with sum 2. In our 1986 paper, referred to hereafter as [DM], we defined the vector space K(/i) of integrals over suitable cycles of n+2

x i )~ >i' d z. i where the integrals are viewed as functions of x (with xn+i = 0 ,x n+2 = l ,a :n+3 = oo).

It is a (n + 1)-dimensional vector space of multivalued

functions on Q, identified with M . In [Lauricella 1893], V(n) is described as the space of solutions of a system of linear partial differential equations expressing all second derivatives in terms of derivatives of order < 1. For g a fixed multivalued function on M of the form U(xi —X j ) a i j

(1 < z < ft, 1 < j < rc 4- 2, i < j )

the m ultivalued functions / •g for / GV(fj.) form again a vector space W of dimension (n -f 1). Such a twist of a V(fi) we call hypergeometric-like . The consideration of all twists of V(p) restores the sym m etry among the indices 1 ,... , n + 3 which was broken by imposing xn+i = 0, xn+2 = 1, ^n+3 = oo: any perm utation 2 of

P 1

. Therefore any holomorphic function on M + extends to

P lS and is constant. P r o o f o f (iv ) Fix a, 6, c £ S and define S i, M i as in §1N. The subspace M i of M m aps isomorphically to Q. It can be identified with the complement in the affine space A Sl of the hyperplanes Xi = 0,ar* = 1 (i £ Si ) and Xi = xj. The closure in A51 of any divisor D of M i is a divisor of A S l , autom atically principal; hence P ic(M i) = Pic(Q) = 0. P r o o f o f (ii)(iii) By (2.4), one has

(3.2.1)

0 — CT(Q)/C* i Z

-» Pic(Q+) -» 0

is exact. For i j j y k y i distinct in S, the cross ratio (ar*,

, a?/) : M + —+ P 1

descends to a function from Q+ to P 1. The inverse image of 0 (resp. oo) is Dik + Dji (resp. Djk -f Du). The cross ratio is an invertible function on Q whose image by v is e,* -f ejt — ejk —eu. Let R be the subgroup of Z generated by these images. Diagram chasing in 0 —►0 * ( Q ) / C *

u

->

R we see th at to prove (ii), (iii) and th a t constants and cross ratios generate G*(Q), it is enough to check:

19

CO M PUTATIONS FO R Q AN D Q+ Lem m a

3.3. For

|S j >

3, the sequence

(s) r

— zV 2 / -I*. z 5 -► Z /( 2) -V 0

is exact. P

roof.

The cokernel of T is generated

by

elements es (s £ S'), with

relations es = —e* for s / t. For s , t , u distinct, this implies es = —e* = eu = —ea. One infers 2e5 = 0 and e5 = et : the cokernel is Z /(2).

H

We prove exactness at Z ' ^ ' by induction on \S\ > 3. If \S\ = 3, R = 0 and T is injective. If \S\ > 4, fix

sq

£ S. If a lies in the kernel of T, one

has Y l aso,t = 0. One can add to a an integral linear combinations of the t

generators of R and replace a by an element in the kernel with aSQ}t = 0 for all t ^ so in 5. This reduces the problem to the case of S — {so} and proves 3.3 by induction. For later use, we mention the corollary: 3.4. Fix s £ S. The Picard group of Q + — { j D s t is Z. All t divisors Dk,t (k,£ ^ s ) are linearly equivalent and their class is a generator C

orollary

of Pic. P

roof.

The Picard group is the quotient of P i c( Q+) = ker(Z5 —►

Z /(2 )) by the subgroup generated by the es + et with s fixed, t varying. The m ap Z s —►Z : a »—►( J 2 at) — as identifies the quotient with 2Z, and tjts 3.4 follows. Applying 2.8, we get COROLLARY

3.5. Multivalued functions on Q of the form f =

with fk E 0 * ( Q ) and

n/r

£ C are uniquely determined, up to a multiplicative

constant, by their valuations vs t ( f ) = EajV, t(fi)- The valuations can be

(f)’

any system of complex numbers v £ C ' z ' f or which

^2

® f or

each s £ S. R em ark s 3.6 (i) The fact th a t the constants and the cross-ratios generate 0 *( Q ) is also easily seen on the model Q ~ M i C A S l : the functions 1 —X( and (Xi — Xj )/ xi are among the cross-ratios.

COM MENSURABILITIES AMONG LATTICES IN P U { \ yn)

20

(ii) The same model shows th at / E 0 * ( Q ) is determ ined up to constants by vafi ( f ) )vt )i ( f ) and V i j ( f ) ( i , j E -Si), and th a t these valuations can be freely chosen. (iii) A more geometric proof of Pi c(Q+) = {a E Z 5 | E as E 2Z} will be given in (3.14).

3.7.

T h e case N = 4 For N = 4, the space Q+ is not separated: the

cross ratio m aps it to P 1, but the points 0, l,o o have to be doubled. Each corresponds to a partition {{s,tf}{u, v}} of S', and the two points above are given respectively by x s = x t and x u = x v . The subspace in 3.4 is ju st a projective line: one point above each of 0, l,o o is deleted. P r o p o s i t i o n 3 .8 .

The class in Pi c(Q+) C Z5 of the maximal exterior

power of Qq+ is (—2, . . . , —2). P r o o f . One has to show th at for each s E S , one has d3(Qg+x) = —2. Fix a , 6,c ^ s in S and define Si = S — { a ,6,c}. Fix distinct x J E P 1 for t ^ s, with x Q a = 0,x^ =

— °°- The ds to be evaluated is a

degree on the projective line x t = X® for t ^ s . This line embeds in Q + — D ab — Dbc — D ca j which can be identified with the open subset of P 51 obtained by removing the intersections, two by two, of the divisors x tl = 0 ,x t2 = 1, Xt 3 = oo and x s = x t . The embedded line is obtained by fixing all coordinates but one in P 51, and since the restriction of the canonical bundle of P Sl to this line is its fi1, * ( 0 5 ? * ) = d e g ( P \ f t 1) = - 2 .

C o r o l l a r y 3 .9 .

The class in Pi c(Q+) of the maximal exterior power

of Og+(log E D m ) is ( N - 3 , . . . , N - 3). P r o o f . The maximal exterior power is

Each degree ds is ds

J +

1 = —2 + (JV — 1) = iV —3.

21

COM PUTATIONS FOR Q AND Q+

3.10

Let S+ be deduced from S by the adjunction of an element w and let

(3.10.1)

tt:Q + (S + )

^ Q + (S)

be the “forgetting x ^ ’-m ap. A point p in Q +(S) is an isomorphism class of systems (P, ( x s)se s ) ’ P a projective line, (x 5) a system of points with at most one confluence x, = x t . The map (3.10.1) is sm ooth, with fiber at p the projective line P if the x s are distinct, and P minus the x s if there is a confluence x s = x t . For C a line bundle on Q + ( 5 + ) , the degree dw is the degree of the restriction of £ to a general fiber of (3.10.1). For s ^ w, the degree ds on Q + ( S ) and Q + ( S + ) is the degree on suitable projective lines. The one for Q +(S) is the image of the one for Q + (5+). The natural inclusion of Z 5 into Z 5+ hence gives rise to a com m utative diagram Pic(Q+(S))

-------- ► P ic(Q + (S + ))

(3.10.2) ker(Z5 -* Z /(2 ) -------- ►ker(Z5 C o r o l l a r y 3.11.

Z /(2)).

The class in P ic ( Q + ( S + ) ) of

^ q + (5 +)/q+(S) 25 9^ven by dw = —2 and ds = 0 for s ^ w. P r o o f . T h e lin e b u n d le in q u e stio n is

ftmax(b of Q + .

3.15

Let \ s,t be the character of G ^ /{ ± 1 } given by the characteristic

function of {5,/} . The corresponding line bundle C on

is characterized

by the existence of a Xs,t equivariant m ap from H+ / S L ( 2) to the total space of £ ; one may take C = 0 ( —D Sft) and the map FSjt. The isomorphism (3.14.2)

of Pic(Q+) with ker(Z‘s —►Z /(2 )) is hence the opposite of the

isomorphism 3.2.

26

COM M ENSURABILITIES AMONG LATTICES IN P t / ( l , n )

Let n G Z (?) ^ / be given. The conditions Y l n *,t = ® f°r existence of t f E 0 * (Q) with valuations n Sti along the D Sjt can be interpreted as E n a *Dat = 0 in Pi c(Q+). On if , there is up to a constant a unique function with valuation n S)t along the divisor hs = ht of i f + . it is F := f l F»,t’* The conditions

(product over ( f ) ) •

= ® can also t invariant by C*5 , hence comes from Q.

interpreted as meaning th a t F is

§4. LAURICELLA’S HYPERGEOMETRIC FUNCTIONS

DEFIN ITIO N

4.1. A local system of holomorphic functions over an an­

alytic variety X is a C -linear subsheaf V of O which is a local system of finite dimensional C-vector spaces. If X is connected, a local section of V near xq E X extends as a mul­ tivalued holomorphic function on X. Conversely, let F be a multivalued holomorphic function, with the property that its branches at xo span a fi­ nite dim ensional complex vector space. By analytic continuation, the same holds everywhere, and the functions on open subsets of X which, locally, are constant coefficient linear combinations of branches of F , form a local system of holomorphic functions. 4.2 If L is a rank one local system of holomorphic functions, the products £v, for £ (resp. t>) a local section of L (resp. V) is again a local system of holom orphic functions, called the twist of V by L. If £ is a non zero m ultivalued section of L, it is also called the twist of V by £. E x a m p le Let }\ be non-vanishing holomorphic functions on X and ^

G C.

The determ inations of / := Ylfk* are constant multiples of one another and k

hence span a rank one local system. The twist of V by this local system is denoted f V . 4.3 We fix a set S with N > 4 elements, a family pi of complex numbers indexed by 5, with S p s — 2 and a, 6, c E S.

We will use the notations

S U M U Q , . . . of JIN. We assume th a t no \xs is an integer. For the integral case, see 4.13. W hen x varies in M\ , the integrals from one ramification point x s to another of a determ ination of cu := Y [ ( z - x s)~*’ dz s^c

28

COM M ENSURABILITIES AMONG LATTICES IN P U( l , n )

span a local system of holomorphic functions on M i. The corresponding lo­ cal system on Q is the local system of Lauricella’s hypergeometric functions, relative to the choices of fi and of a, 6, c. 4.4 We put together from [DM] 9.5, 4.6 the following results: (a) The local system of Lauricella’s hypergeometric functions is of rank N — 2. If Fi (0 < i < N — 3) are linearly independent local sections, the F( are the projective coordinates of an etale map from Q to p

N ~ 3

(cf. 6.3

below). (b) Fix s G Si and t £ S , t /

c.

Assume th a t fis + fit £ Z.

Fix

q G D Sit C Q + and a small ball B around q. On B C\Q, one can find a m ultivalued basis F{ (0 < i < N —3) of the Lauricella’s local system, i.e., a basis on the universal covering of B 0 Q, with the following properties. (a ) F i , . . . , Fjv-3 are univalued and extend holomorphically across D S)t ; (/?) For z a local equation of D S)t,Fo is of the form

(holomorphic

invertible on B). (7 ) Let 7r : S —> S st be the quotient of S obtained by identifying s and t and define p, on S st by pi =

^2 Pj ' Pir(a) = Pa for a ^ s , t 7r(j)=* = n s + fjLt . We identify D S)t with Q (S st ). On D Sjt = Q ( S st),

and

the Fi (1 < i < N — 3) span the Lauricella hypergeometric local system, relative to 7r(a), 7r(6), 7r(c) and p. 4.5 We keep identifying Q with Mi as in 4.3. Fix 0! ,6 ',c' E S and let (p be the projectivity mapping (xa/, x*,/, x c>) to (0, l,o o ). It is the cross-ratio map z s Z XX] ) > . The fiber space can be viewed as being Q (S + )-> Q (S ) (identify Q (S+) with M \ (S+)).

Locally on Q + (S+), the form u> is the

product of a non-vanishing section of ^ g + (s +) /Q + (S') by a product fl/fc * k with fk G (D*(Q(S+)) and fik G C. It has a valuation Vi,t(u>) along each D S)t(sj t G 5+). Along D 8fW the valuation of the form u of 4.3 is —fis . Along D s t with s , < G S , s G S i , t ^ c , the valuation is 0; it coincides with the exponent a S}t of V (4.8). Let / be a multivalued function on Q of the form [ j f £ k with /* £ 0 * ( Q ) and ctk G C. The twisted local system f V is obtained by integrating the multivalued form r) = f w on chains in the fibers. For s , t G 5, s G S i , t ^ c the exponent a Sit( f V ) = vsj ( f ) + a $it(V) = vStt( f ) is equal to the valuation vStt ( f u ) = vS)t( f ) of fuj along D 8yt C Q + (S'+ ). If a, 6, c are changed, the role of u is played by another relative multival­ ued 1-form u/, and f u = / 'u / , the m ultiple of u) 1 (4.5) whose integration yields f V . T h at is, independently of the choice of a, 6, c, a hypergeometriclike local system V relative to /i (4.7) is obtained by integrating a m ulti­ valued form r) on the chains in the fibers. For each choice of a, b, c, 77 can be w ritten as fu; and

32

COM MENSURABILITIES AMONG LATTICES IN P U( l , n )

for s, t G S', s G S i, t ^ c. As a , 6,c are freely chosen, the equality persists for all s , 2. One also has ,W(??) —*

PS •

Let V be the unique connection on ^ q ( 5+)/q(s) for which, locally, the determ inations of the multivalued relative 1-form rj are horizontal. Such because the determ inations of rj do not vanish and

a connection exists any

two differ by a multiplicative constant.

The line bundle ^ q (s +)/ q (s )

is the restriction to Q(S+) of &q +(s +)/ q +(s ) anc^ connection V has a lograithmic pole along D S}ty with residue (asdefined in 2.9)given by Res(V) = —v 8 j(r)). Proposition 2.10 gives c l( ^ Q + ( S + / Q + (S )) =

^ s , t ( ^ l ) c ^ ( D 8 }t)

in H 2 (Q+(S+), C) as well as in Pic(Q +(S+)) 0 C by 2.6. Applying 3.11, we get 2

\ /

W».‘W = i

0

* 7* w

if o

^ " 2

s = w‘

lf

This gives - fis =

0

t£ S

(i.e., 4.9.1) and = - 2. te s

4.1 3 4.3

Following [DM] 2.15, we now explain how to modify the definitions and 4.10 when some /i5 are integers. This will not really be needed in

our main results, but it does allow for neater statem ents. We will be brief and we do not discuss here what happens along D Sjt when 1 —fis — fit G Z. We identify Q with M i. On P 1 x M i/M i, we have sections x $ (s G S). Let L be the local system on the complement of these sections such th at (a) L is trivialized on the subset {(z,m ) | z real, su p |x 5| < z < oo} s^c

by a trivialization e; (b) on each fiber P 1 - {x s( m ) | s E S ) , L has monodromy a s = exp(27ri fis ) around x s(m).

33

LAURICELLA’S H YPERG EO M ETRIC FU N C TIO N S

The product u :=

—£a)-/i# ' e ' dz s^c

extends as a L-valued relative 1-form. If

is an integer, L extends across

the section x s; if further fis < 0, u> extends; as a relative 1-form over M i, its restriction to x s( M \) is zero. Let 5 ' C S contain all s with

an integer

< o, and no s with iis an integer > 0. Let P 9 be the fiber space P x M i over M i minus the sections x s , s £ S '; let 7r' : P ' —+ M\ and let V be the extension by 0 of L to P 9. The relative 1-form lj has a class [a;] which is a section over M \ of ( R 1 tt'L ') 0 c Omi* P u t S " = S —S'; define 7r" : P " —►M i and L" by exchanging the roles of S 9 and 5 " . The dual of the local system R xir' 1/ on M i is the local system R 1 7r"(Lv )", for Lv the local system dual to L. One can view [a;] as a map

[ w ] : R 1 x 9 :( L }' ) " ^ O m 1The image of this map is Lauricella’s local system of holomorphic functions on M i. It does not depend on the choice of S 9 as above. In more concrete terms, it is the following local system of integrals. One takes the C-linear span of the (a) integral from x s to x t) when neither /i5 nor

is an integer > 0.

(b) residue at x s , when fis is an integer > 0. The properties 4.4 (a)(b) continue to hold. The local system obtained varies holomorphically with (i. If the decomposition S = S ' II S" is kept fixed, this is clear, and if a decomposition S' II S " is allowable for one //, it remains allowable for nearby jj. E x a m p le 4 .1 4 Take fis = 0 for s G S — {c}, fic = 2. The corresponding local system is spanned by the functions 1 and

(s E S i).

E x a m p le 4.15 Let us explain the continuity in fis when /z5 becomes an integer > 0. The integral from x t to x s is first reinterpreted as an integral along a cycle T with coefficients in the dual local system Lv , as follows (cf. [DM], 2.6)

34

COM M ENSURABILITIES AMONG LATTICES IN P C 7(l,n )

/ a section of Lv near y. The monodromy around x s is responsible for the factor a j 1. The condition th at T := T' + A r" be a cycle is: 1 = A —X a J 1. We let / vary holomorphically with /i. The cycle (I — a J X)T tends to the circuit T" (with constant coefficients around x s) when a s —►1, and the corresponding integral tends to a residue.

4.1 6 The monodromy of a hypergeometric-like local system on Q is a linear representation of 7Ti(Q). The corresponding projective representations are the ones considered in [DM], for the same \x.

§5. GELFAND’S DESCRIPTION OF HYPERGEOMETRIC FUNCTIONS In their theory, Gelfand et al. consider local systems of holomorphic func­ tions on a suitable Zariski open subset of H o m ( C p , C " ) . For p = 2, it is an avatar of Lauricella’s, and we will use the pleasingly simple expression they get for the differential equations which are satisfied. We prefer using C 5 to C N . W ith this change in notations, for

5.1.

p — 2, the local system they consider is on the Zariski open subset H C H — H o m ( C 2 , C s ) (3.12); it depends on the choice of complex num bers p s (s 6 5) with Dp, = p = 2. We assume at first th at no

is an integer.

For the general case, see 5.5. We continue the notations of (3.12). On C*5 , one fixes the multivalued function . Its pullback by h £ H is the multivalued function

:=

on C 2 minus the lines h s — 0. Let E be E uler’s vector field z\d\ -I- zo