Holomorphic Hilbert Modular Forms 0534103448, 9780534103446

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Holomorphic Hilbert Modular Forms

Paul B. Gar.~tt University o_f Minnesota

Wadsworth & Brooks/Cole Advanced Books & Software Pacific Grove, California

Brooks/Cole Publitbina Company A Division of Wadsworth, Inc. © 1990 by Wadsworth, Inc., Belmont, California 94002 All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transcribed, in any form or by any means-electronic, mechanical, photocopying, recording~or otherwise-without the prior written permission of the publisher, Brooks/Cole Publishing Company, Pacific Grove, Califomia 93950, a division of Wadsworth, Inc.

Printed in the Un1tec! States of America 10 9 8 7 6 5 4 3 2 1 Library of Congres.1 Cataloging in Publication Data Garrett, Paul B. Holomorphic Hilbert modular forms.

Bibliography: p. Includes index. 1. Hilbert modular surfaces.

I. Title. QA573.037

1989

516.3'5

2. Holom.orphic mappings.

88-33753

ISBN 0-534-10344-8

Spons9fing Editor: John Kimmel

~ar\e~ Representative: Cathy Twiss E~itorial ~uistant: Jennifer Greenwood Pr~uct-on Editor. Linda Loba Manuscript Editor. Anne Draus Permissions Editor: Carline Haga Interior and Cover Design: Vernon T. Boes Typpsetti~g: Asco Trade Typesetting Ltd. Cov~r Printing: Phoenix Color Corporation Printing and ~inding: Arcata Graphics•Fairfield

Introduction •

This book is intended as a systematic, self-contained, and straightforward introduction to a substantial part of the theory of holomorphic Hilbert modular fot 111s, associated L-functions, and especially their arithmetic. As such, it is an introduction to the theory of automorphic forms in general, especially the arithmetic of l1olomorphic automorphic forms. Beginning from the most standard algebraic number theory and function theory, one can.develop matters far enough to recover some of Shimura's results on special values of L-functions attached to Hilbert modular forms. Also within reach are the theorem on special values of L-functions of totally real number fields (after Klingen), and examples of results on special values of certain grossencharacter L-functions (due to Damerell over Q and to Shimura in general). One of many reasons for the study of Hilbert modular fon11s is for applications to Dirichlet series. One emphasis of this book is the construction of Dirichlet series

v1hich have analytic continuations and functional equations. Many such are obtained as (Mellin) integral transforms of holomorphic Hilbert modular forms: the analytic continuations and functional equations follow almost immediately from the properties of such modular forms. Further, the existence of an Euler product facto riza ti on D(s)

=

n

l /Pp(p-s)

p prime

(where PP is a polynomial) of a Dirichlet series arising from a modular form f is essentially equivalent to properties off: f should be an eigenfunction for certain operators (the Hecke operators) on automorphic for111s. · The first three chapters bear upon such matters. A considerably subtler issue is the deter1nination of the nature of the values of such Dirichlet series D(s) at special integers s. Some samples of results in this ~irection are given in sections 6.2- 6.5. The discussion of the arithmetic of the special values of these Dirichlet series relies profoundly upon arithmetic properties of modular for111s themselves.

·

,,;

Introduction .

I~ is possible to give a reasonably full accot1nt of these speciai-values results by virtue of new direct proofs (chapter 7) of a theoren1 (stated ,in \6.1) of which all these special values results are corollaries. One half of this theorepl is part of a result of Shimura concerning the arithmetic properties of Fourier coefficients of Hilbert modular for111s. The other half is an apparently new comparison of inner products of cuspforms, that allows an omission of some delicate matters concerning Hecke operators. In the first five chapters I have attempted to collect in one place fundamental ideas and 1nethods that arose during the period 1904-1961. The diffuseness of the literature and the desirability of illustrating the efficacy of''modern methods'' I have taker1 as justification of this reprise. Chapters 6 and 7 bear upon more recent (and more arithmetic) d_e velopments. All but the method of 7.9-7.10 and the choice of prese11tation must be attributed to many other authors-notably, Blumenthal, Hecke, Siegel, Maass, Rankin, Klingen, Shimura, and Selberg. Tl1ere is no presumption of familiarity with the theory.of elliptic functions and elliptic modular fo1111s, (although some experience might prove helpful). Indeed, the beautiful but very special ideas and calculations arising from the theory of elliptic functions may not be the best paradigm for general ex~tations: there is no convenient generalization of the Weierstrass P-function. There are even more mundane coincidences that are implicitly exploited in the classical theory of elliptic modular forms; for example, that the rational integers are a Euclidean (hence principal) domain. But, as it turns out there is no genuine need to depend on such fortuitous coincidences, and because the general situation is important in its own right, I have tried to give a portrayal of Hilbert modular forms that is essentially blind to special features of the ground field. It is possible to do so coherently and • palatably by consideri11g modular f01 rns on adele groups. • Currently, it may seem that the theory of elliptic modular fo11ns (that is, Hilbert modular for111s with ground-field Q) is most immediately relevant to other parts of arithmetic. However, Klingen's deter111ination of the arithmetic nature of the special values of L-functions of totally real number fields (see 6.2) is a pointed example of the necessity and utility of more general considerations. Also, the treatment here (6.3) of Damerell's and Shimura's results on grossencharacter Lfunctions makes essential use of the arithmetic properties of Fourier coefficients of Hilbert modular forms (due to Shimura). (See 6.1.) Further, the arguments of 6.3-6.5 rely u·pon comparisons of inner pr0ducts of automorphic forn1s; 6.1 gives a new result in this direction. The new proof (Chapt~r 7), which yields a part of Shimura's theorem on the arithmetic properties of the Fourier coefficients of Hilbert modular forms and gives the comparison of inn~r products, suggests another broadening of perspective and methods: the discernible arithmetic of 'Certain (Siegel's) Eisenstein series on a larger group (a symplectic group of rank 2) is the starting point of the proof. Thus, Siegel's already striking idea (in [Si2]) for understanding the arithmetic natur.e of the Fourier coefficients of more genf;ra.J Eisenstein series becomes more significant. Shimura's original proof (in [Sh3]) c,f the theorem on Fourier coefficients depends on a special case of his profound results (in [S111]) on canonical modets of arithmetic quotients, that in tum depend on quite serious results from algebraic geometry.

lntrodul·tion --

While the method of7.9 arid 7.10 seems incapable of recovering Shimura's theorems generation of class fields, it is profitable to see that the arithmetic nature of the F'ourier coefficients can be understood directly. Previous methods of comparison of inner products of cuspforms relied essentially on the delicate notion of''newfor111'' (as in [M], [A], [CJ); the idea here is less technical and is relevant in more general siiuations as well. Another point hopefully illustrated here is the central role of Eisenstein series -- in Rankin's integral representation of L-functions (4.10), in Shimura's applications (6.3 - 6.5), and in the present proof of the theorem of 6.1 (Chapter 7). The first chapter studies Hilbert modular f01111s from a ''classical~' viewpointthat is, as it would have been done prior to 1960. Siegel's book [Sil] also gives an ir1troduction to this material. Most of this material merely imitates the theory of elJiptic modular fo1111s. An exception is the method of proof of the finitedimensionality of spaces of cuspforms in 1.7; this method is due to Siegel and Maass. This method, applicable as well to elliptic modular for111s, supplants the use in that theory of either residues or the Riemann-Roch tl1eorem, neither of which is as helpful in general. Precise attribution of all historic sources is beyond my ability; the results t1ere are due to many authors, going back as far as Blumenthal's pursuit of Hilbert's suggestion in 1904. Some of the more important sources are [Bl], [B2], [I-I I], [H2], [H3], [KLl], [M], [P], and [Se]. With regard to some matters (Hecke operators and L-·f unctions) the treatment in this chapter is only introductory, as these topics are best treated later in a different way. For the classical versions of these topics-that is, where the base field is Q, one might consult [Sh2] or [Gu]. The second chapter introduces a general notion of automorphic form on GL(2) over the adeles. The influential books [GGP] and [JLJ make the usefulness · of this viewpoint clear. This viewpoint makes any ground field tractable by unifor1n methods. For example, a classical treatment of Hecke operators for congruence subgroups over rings of integers not of narrow class number one is excruciatingly technical and not very helpful; an adelic treatment supplants this approach by a discussion even simpler than consideration of Hecke operators for SL(2, I ). ·F urthtr, in this setting the Euler product expansion of Dirichlet series obtained as Mellin transfo rms of modular forms arises as inexorably as in Tate's thesis's treatment of L-functions of number fields (as in [La] or [CF]). In 2.7 some references to recent de vel o p1nents regarding analytic continuation of higher L-functions are given. . .1\ppendices A.1, ;,.2, and .4..4 give some relevant background en integration theory o n homogeneoLs spaces, har111onic analysis on the adeles, and invariant differential 011

.

operators on SL(2, IR). Both for completeness and for clarity, the third chapter compares the classical and adeJic versions of these things. An essential ingredient is a Strong Approximatiol1 'fh eorem for SL(2); this theorem is proven in Appendix A.3, which specializes and simplifies a proof found in [Kn]. Although it is not hard to give a qt1ite elen1cntary proof over (Q (for example, in Chapter 3 of [Sh2] ), it is not easy to do so over arbitrary number fields; the original references for the ideas i11volved in even more general strong-approximation tl1eorems are the papers [El] a11d [E2] . I do not go far in discussing Hecke operators at bad primes; for such a discussion, see [A], [M], and [CJ.

Mi

Introduction

Chapter 4 is concerned with Eisenstein series. Consideration of analytic and arithmetic properties of Eisenstein series goes back almost 200 years and is still central to number theory and automorphic for1ns. Most of this chapter is.. a study of the formal properties of Eisenstein ~eries, which become even more transparent in an adelic formulation. The method of proof of analytic continuation and functional equation given in 4.2 refor111ulates an idea I first saw in Godement's exceptional article [Go]; perhaps the idea should be attributed to Maass and Selberg in some for 111. This avoids direct con,s i'ieration of confluent hypergeometric functions and their p-adic analogues, although the latter have virtues of their own. Section 4.10 is a mildly adelic \iersion of a part of Rankin's papers, [R 1] and [R2]. (See also [P].) Although Rankin~s first paper on these matters appeared in the 1930's, the potential of the ~eneral idea is not yet exhausted. Sections 6.3-6.5 pursue the arithmetic side of the jdea (decisively illustrated in Shimura's papers [Sh4],'[Sh5], and [Sh6]). In Chapter 5 the treatment is again somewhat classical,.for· a -non-t:rivial . . . reason: theta series that are of half-integral weight cannot be considered as automorphic forms on an adelic GL(2); indeed, the study of half-integral-weight ·automorphic for111s.js a subject in its own right and is not pursued here. The main point of this chapter is the proof that theta series are modular forms; the proof uses an adelic version of the Poisson summation for111ula. Theta series have been studied for a long time; some references are [B2], [KL2], and [Sc]. Theta series of integral weight reappear in 6.3 in connection ,vith grossencharacter.-.L -functions. Chapter 6 discusses so1ne developments since 1961. Section 6.1 states part of a fundamental theorem of Shimura~s (from [Sh3]); the theorem concerns the arithmetic properties of Fourier coefficients of Hilbert modular for1ns. I cannot believe that this result was not known until 1975, considering that version~ of this result over Q were known before 1900 via the theory of elliptic functions. (Certainly the other parts of the results of [Sh3], those regarding generation of class fields, could not have bee11 known earlier.) Also, the theorem stated in 6.1 contains an equally important fact concerning comparison of inner products of cuspfor111s; this fact is indispensable for applications. Section 6.2 recapitulates Klingen's papers [Kt] and [K2]; that is, it combines the result of 6. l over Q with 4.8's calculation of Fourier coefficients of Eisenstein series to determine the arithmetic natt1re of special values of certain L-functions of totally real number fields. Sections 6.3-6.5 give a substantial class of examples of Shimura's results on special values of L-functions attached to Hilbert modular cuspforms ([Sh4], [Sh5], and [Sh6]). The results of 6.5 for ground-field O were obtained in a different way in [DJ. All these special-value results fit into the general conjectural patter11 enunciated in [De]. The last chapter is devoted to a new proof of a part of Shimura's theorem (6.1) on Fourier coefficients and to a proof of the comparison results concerning inner products. Sections 7.1 - 7.8 follow Siegel's idea (in [Si2]) to prove the rationality of Fourier coefficients of some specific Eisenstein series on a symplectic group of rank 2. Such calculations can be done in much greater generality, as in [Sh7], for example, Section 7.9, following the idea of [G 1], determines the restriction of this Eisenstein series to an in1bedded copy of SL(2) x SL(2), embodied in the ''Main Formula." Section 7.10 combines these two items to obtain the theorem of 6.1.

Introduction

•

IX

•

As compleme11tary and supplernentary reading, one might c0nsult the books [Sh2], [Sit], [GGPJ, [JLJ, [Ge], a nd [Gu] . I have omitted ma ny topics and applications, most of wh.ich would l1a vc the subject matter (.1f tl1is voit1me as prerequisite: algebraic geiJrr1etry of lfilb~rt modular varieties . con1pt1ctifications of arithmetic quotients~ m,Jd u] i of abelian varieties, generation _of class fields, base change, no11-h·olo1norpl1ic modular forms, l1alf-integral-weigl1t forms, converse theorems, representation lheor1·, trace formulas, Kronecker lirnit formulas, congrue.nce properties of mc>dt1lar for1r1s~ application to Hasse-Weil zeta functions, Siegel-Weil formuJa ", tl1et.,1 C() rrcsponden•ces, triple-product L-f11nctions, .... ! find it i11teresting to trace the ,1rgume11ts used here to obtain the rationality results. The arithmetic of Dirichle t L-ft1nctit).r1s over Q is quite old; ir1 appendix A.5 tl1e results are proven by residues an

JJ

u u

the Main F'orn1 ula (7.9 ), methods of 7.10 ~

ratior1ality properties for GL(2, 0) (6.1 over 0)

u U u

Eisenstein series Fourier expansion (4.8) ~

Klinge_n 's method (6.2)

~

~~

i

~~

ij

special valLtes of Dirichlet L-functi ()ns over totally real number fields (6.2) U •

u u u u

.

Siegel's 1nethod (7 .1 -- 7.8) ~

rationality of Fotirier coefficients (>f Siegel's Eisenstei11 series over totally

U

real fields ·,

the Main Formula (7.9)1 methods of 7.10 ~

u u u ~

rationality properties for G1.,(2) over totally real fields (6.1)

x

•

Introduction

Rankin-Shimura metht)d f4. l 0, 6.3--6.5) ~

special values of grosse11cl1aracter L-functions over totally imaginary quadratic extensions of tt1tally real number fields, special values of standard L-functions and convolution L-functions for GL(2) •

'

4

If one views the theory of Hilbert modu!~1r forms as but the second example (after elliptic modular fo11ns) of mucf1 inore general phenomena, the~ distilling the facts and finding the most economical vie,\.'P()ir1t becomes important: a long 1v·oyage witl1 hea"'Y baggage is unplca~ant. P,trtl)i for this reason, I l1ave attempted to be conservative in cl1oosing what to include: almost everything in tl1is book is indispensable for get ting to the rest1lts of C:hapter 6. The 11e\v methods of 7.9-7.10 allow all these n1atter8 to fit into one volume, most of which ·is prerequisite for a reasonable understanding of further developments in tl1e subject. Give11 the modest prert~quisites, tl1is b(1ok ought to be accessible to graduate students and non-specialists. I also hope that it will be useful as a handbook of sorne standard metl1ods in the subject . I was introduced to this subject thro~gh Shimura,s lectures at Princeton Universit}· from 1975 to 1977. The text is a rewriting c.1f parts of courses and lectures I have give11 at tl1e l.Jni\'ersity of !\1innesota from 1982 t.o 1988. While almost all these results can be fot1nd somewhere in the literatt1re, tl1e precise forn1 of this text is the rest1lt a. -t (z -

( z, /J) a).

Now it must be that

..

M=m y ~

for some ideals nt and m', and for some y, y' E F 2 (from the structure theorem fot finitely generated modules over Dedekind domains). Now /\2 F 2 is one-dimensional, so a I\ y = x(a ' I\ y' ) for son1e x E Fx. Replace y' by x - 1 y'·and m' by xm', so that now r

ao 2

= oo: ' EB my'.

Take A E GL(2, F) so that Aa = a' and Ay = y'. Since et " y = r:t' /\ y', A E SL(2, F). Since A and A- 1 send ao 2 to itself, A and A - 1 preserve o 2 . So A E GL(2, o). Thus, we have A E SL(2, o). ■ -

COkOLLARY For any congruence subgroup r, there are only finitely many r-irteqnivalent cusps. Let {K 1 , . . . , K..,} be a set of irredundant representatives for ·.the inequivalent cusps. Let bi E SL(2, F) so that c5;(ioo) = "i• Let P be the group of upper triangular matrices in G L + (2, F). Then

r-

(disjoint union). i

Proof Let r(n) be a congruence subgroup contained in Z(o)r, and let v be the index of r(n) in GL +(2, o). Then there are at most vh(o)r(n)-inequivalent cusps of r(n); therefore, the number of inequivalent cusps of r is no greater than vh(o), which is finite. · It is immedi~te that Pis exactly the isotropy grou-p of ioo in GL +(2, F). Given g e GL +(2, F), suppose that g(ioo) = K, where " is some cusp; Let ye r so that y(K) = Ki, for one of the "i• Then (t5;- 1}'g)(ioo) = ioo ; therefore, J,- 1 yg e P, so g e y- 1 c5;P. It is clear that bi is uniquely determined. ·• .

COROLLARY Let f be a weak modular f 01111 of weigbt le with respra to some congruence subgroup r. Forge GL +(2, F), let the Fourier expansion of fltg be

.!

SECTION 1.4 Koecher's Principle

(fltg)(z)

9

= L c~(g) exp(2ni Tr({z)). ~

.

. Let {hi}

be a (finite) set of representatives for

For f to be ~qrspform, it is necessary and sufficient that c~(g) positive Merely for g E {bi}.

= 0 for { not totally

Re1111Uks _ The point of this assertion is that there are only a tinite number of g .E GL +(2, F) for which the condition on Fourier coefficients need be verified .

Proof

For ye r, fltY

•

= f, by definition. On the other hand, for

a b p= 0 d EP ""

and for a Hilbert modular form . . O

n~O

Since 2

Tr(11 "e)

= 11J"~j +

L 11?"e;,

i#j

from crude estimates we see that (no matter what weight k is) the subseries cannot be convergent unless actually c(~) = 0. The Fourier series for f(z) is absolutely convergent for z e f)'" (since /(x + iy) is smooth as a function of x), so we find that

SECTION 1.5 Holomorphic Eisenstein Series of Level One .

c(~) = 0 unless every ai(~) is non-negative; therefore, either positive.

11

.

e= 0 or ~ ts totally ■

Proof of Corollary_

We can apply the theorem to fltg for any g E GL +(2, F), s~nce we have already seen that fl 1 g is in Wfm(k) if f is. That is, for all g EGL +(2, F), f l1cg . has non-zero Fourier coefficients c~(g) only for = 0 or totally positive; f is a. Hilbert modular form. A similar argument applies to the case where not all ki . are the · same; all weak Hilbert modular forn1s of such weights are necessarily cuspforms. ■

e

1.5 Holomor,phic Eisenstein Series of Level One The simplest examples of Hilbert modular forms are the holomorphic Eisenstein series. We will consider just Eisenstein series for GL +(2, o) at the moment, though an a~alogous construction ,vorks mucl1 more generally. Eisenstein series ·o ften appear in two guises, referred to here as ''homogeneous'' and ''inhomogeneous.'' While the spaces spanned by these two versions of Eisenstein series are the same, the two sorts of Eisenstein series arise and are useful in differept circumstances. So it is important to appreciate that they span the same space of functions. · In this section we will compare the space spanned by the two types of Eisenstein series and deter111ine bases for this space. Fµrther, we will see how an Eisenstein series may be associated to each cusp of 6L +(2, o), in such a manner as to dete1 n1ine a basis for the whole space spanned by all the Eisenstein series. Also, . we will show that, with weight 2k = (2k, ... , 2k), . •

=Cfm(GL +(2, o),2k)(f){space spanned·by Eisenstein series of weight 2k}. .

.

~hat is, every modular fo1111 of such weight for GL +(2,9) is a cuspform plus a .linear combination of Eisenstein series. (We have already noted that if the weight has any two components unequal, then every holomorphic -~ilbert modular form is a .cuspf0·11n). • .

Definition

The inhomogeneous holomorphic_Eisenstein series of weight

2k = (2k, ... , 2k)

>

.

,

with 2 < 2k ~ 7L., associated to a non-zero fractional ideal m of F, is defined by E(z; m, 2k)

=

L*

(cz

+ d)- 2",

•

c.d~m

using a multi-index notation, where

(c, d) e {(ex, fl) E m x m: exo

L • indicates the sum over

+ po = m} /ox.

12

CHA Pl.ER 1 (,"'/assical Theory of Hilbert /~1odular Forn1s

T'he ho1noye11eous liolomorphic Eiserzstein series of weight 2k, associated to a nonzero fractional ideal n1 of F , is defined by E;;(z~ n1, 2k)

L'

=

(cz

+ d)-ik )

c,de m

using a multi-index notation, \\·here

I ' indicates the sum over

(c, d) E ((rn x nt) - {O} )/ ox.

PROPOSITION The series defining the Eisenstein series E'(z ; nt, 2k) are absolutely convergent if a11d only if 2k > 2 (and then uniform I)'· so, for z in compact subsets of ~n'). T11c series defining the Eisenstein series E(z; tit, 2k) are abso]utely con\'ergent for 21< > 2. Pr(>o_f First, 11ote that the series for E(z; 111, 2k) is a su lJseries of that for EJ(z ; 111, 2/f a suitable linear combination E of E.isens tein series from _f m 0, with Fourier coefficients en (0 < n El). and suppose that c" = 0 for n < N. The function 12 yk IJ.(z) l is SL(2~ ~)-invariant; therefore,

sup yt 12 IJ.(z)I f)

==

sup s

}'1i:12

1J(z~.

Since f is of exponential decay on S, the latter supremum is attained, say at z 0 = x 0 + iy 0 E S. On the other han4, we also have an estimate C

= sup exp(nNy)lf(z)I < s

oo,

and this supremum is also attained, say at z 1 = x 1 Fork < 0, for a11y z = x + iy in ~ we have

As k < 0, the last expression is decreasing as y 0

~

+ iy 1 .

oo and so is dominated by tt1e

25

SECTION 1.7 Finfte Dimensionality of Spaces o.f Cuspforms. Estimates on Cuspforms

expression obtained by replacing Yo by}:

Then, for all y > 0, we have

lc"I = exp(2rrny)

+ iy)exp(-2ninx)dx s

f(x [0 . 1)

s

exp(2nny) sup

1/(x + iy)I s

exp(2nny)y- k12 C(½)112 exp( -nN / 2).

O~x ~ l

Since this holds for all y > 0, we may let y ➔ o+ and find that c" = 0: Cfm(r, k) = {O}. If there is cp E Mfm(r, k) with k < 0, then let 1/, be any (not identically zero) • cuspfo1111 (of positive weight) with respect to any congruence subgroup. Then 0) for any non-zero cuspform f E Cfm(r', k ' ), the elements N

It is clear that tl1e su m is a decreasir1g function of .}' 1 as y 1

occurs for 1Vk

12

)' i

- . ctJ,

so its maximu1.t

= 1; therefore,

< (2k)k 12

L

exp(rrn/ 2) exp( - n(n -

i lv)) ==

"> ;v

== (2k)k 12

L n>N

exp(

-1 n(11 -

N)) == (2k)k 12

L exp(-~ rrn). n>l

The righthand sitte is bounded independently of N, so N must be bounded.

Now if Cfm(SL(2, Z), k) were not finjte-dimensional, we could find a cuspform witl1 arbitrarily many F ot1rier coefficients vanishing, contradicting the previous result. Tl1is gives the result for 1~L(2, Z). ■ For the general case, we rnust deal witl1 various complications, tnost of which are purely technical rather th,1n concepttial. An example of this is provided by tl1e third assertion of the len1n1a belo\v, \\'l1ich has a trivial a11alogue form == 1. First consider k == 0: Let

0, for an)' not ide11tically zero cuspforrr1f E Cfm(1', k' ), the ele1ner1ts O, ,y« ~ (a/ecr exp(cy). iii. For y 1 , ••. , y,.. > 0,. . '

•

•

::: mn11m

1/m

fl Y;

•

i '

.,

••

.

Proof The first.assertion follows from calculus; the second is merely a paraphrase of the first. The third asscrtion.follows··from application of the ""Lagrange multiplier n1ethod": It is clear th_a t the infi1P':lin must occur on the hypcrsurface fli ~' :m: n. Tl1e · infimum -will occur at a point (c; 11 • •• ·, ,.}so thaf (for some A E lR) ..

•

That is, • •

•

..

-

F rom this, A. =-= n/{,y. for all indices i; ..we find that . ..

Tberef~ore, at 1t1e infimum, /

nm-1 I

I

fly i

· 11m

)

)L l ' ·

·f

= nl /m

11 }\ •

'

l / 11t /

I .V;,

1/m

II

•

Let {i 0, Re(z) in a fixed compact set}. For each index i, and for f

E

Cfrn(r, k), let

CHAPTER 1 Classical Theory of Hilbert Modular Forms

•

Also, write n occurs. Since also~ E Aj, there is E > 0 so that N ~ whe.never > 11. From the lemma above, for any such ( >

N,

mnlfm

n yflm = (n/N~) l/m

X

n yf fm

1n(N~)lfm

i

~ (n/ N ' )lfm X

Tr~}'
0, i.e., if and only if kJ > 2. This proves the asserted co,nvergence. Assuming convergence, it is purely formal that these Poincare series are in Mfm(r, k). What remains to be shown is that the Poincare series are c1Jspforms . . Applyir1g a criterion we have used before in discussion of Eisenstein series, we need sl1ow ()nly that lim (f lkg) (il) ;. -

=0

00

for every g

E

GL + (2, .f.,). From the definitio11,

(fl"g)(il)

= L µ(y, g(i.,l))-k exp(2ni Tr(v(yg(iJ))))µ(g, iJ) - k = y

~

L µ(yg, iA)-k exp(2ni Tr(v(yg(il)))). y

As ). ~ oo, µ(yg, il)-k goes to O unless yg is upper triangular. In case yg is up 1~r triangular, the exponential term goes to O as .A. ~ oo. In either case the decay is monotone, so by the Monotone Convergence Theorem the sum goes to 0. Tl1at is, the Poincare series f is indeed a cuspform. ■

PROPOSITION f(z)

Let

= L c~exp(2niTr(~z)) 2, and let P(z; k, v) the v-th Poincare series of weight k for r. Then 2, this series is absolutely convergent, and uniformly so for (z, w) in compact subsets of .5m x bm. As a function of z, Q(z, w) E Cfm(f, k) for each fixed w E 5m, and is a cuspform in w, with respect to the congruence subgroup r~, where b q

a

d b C a

d

C

f~urther, for f

E

•

Cfm(r, k),

fl

(f,Q(•, -w)) =

j= 1 ... . • m

{(2ni)(-4i) 1- k 1 f(k1 - 1)} x /(w).

Again we use the idea that L 1 convergence (on compacta) of a sequence of holomorphic functions implies uniform pointwise convergence (on compacta). Note that

Proof

L

yk/2

lµ(}',Z)-k(yz

+ w)-kl

yer

is r-invariant. It suffices to show that

By unwinding this integral (using properties of integration on homogeneous spaces), it becomes

Since everything is non-negative, this unwinding is justified. This integral is a product of the m integrals

~

y" 112 l(z + w1)-11 1.v- 2 dx dy.

Let wi = u integral is

+ iv E .f,.

Replacing x by x - u, and then replacing x by x(y

+ v),

this

56

CHAPTER I Classical Theory of lfilbert J-,,odu/ar Forms

(x2

+

y k j /2 ( y

x

l)-kii2 dx

R

+ V) l - k J/2 y- 2 dy.

(0. oo )

Elementary and well-known estimates give the cor1vergence of these integrals for kj > 2. For the reproducing property, a similar unwinding gives

One may justify integrating term wise in the Fourier expansion off; the effect on an exponential is exp(2rci Tr ~z)yk(z - \•v) - 1c y - 2 d.x d y,

exp(2ni Tr ~z)-. S,'"

which factors into m integrals ,..

exp (21t i ej z) y k ( .x - i }' -

wj ) - k

.v- 2 dx dy.

~

Integrating first in x

E

IR by residues, this becomes

2ni ~

(0 ,oo )

= (2n i) (2ni ~i )k

j -

i

exp (2n i ~1w1)

= (2ni)(21ti,J)"1 - 1 exp(2ni~jwj)(4nej)t-kj r(kj = (2ni)(.-4i) 1 -kJf(ki

1)

=

- l)exp(2ni~iwi).

This pro, es the reproducing pro pert y. To show that Q(z, w) is a cuspfo rm in z, rewrite Q(z, w) as 1

L

Q(z , w) =

I

µ (J}'. z)- k(byz •f- w )-k

yf;r x-\ r ~t: r :r

11

0

X

~

1

: .x in a lattice ;"\ in F

C

[Rm

depending on

r

..

is the subgroup of r consistin g of upper triangular unipotent matrices. As

SECTION 1. I 5 Heck e Operators in a Special Case

µ(oy, z)

= µ(~, yz)µ(y, z) =

57

µ(y, z),

this is

•

L

, er \ r

L

µ()', z)- lc

, eroo\ r

(yz

+ w + A)-k =

;., e "

co

=

L

µ(y, z)-k

L n {(- 2ni)ki - l f(kj-1)} ~

~k- l

exp(2ni Tr

This is the n-th Hecke operator. 1.. his is well-defined (if convergent) since for }' and b E ~(n) we have

E

r

PROPOSITION The Hecke operators map Mfm(r, k) to Mfm(r, k) and Cfm(r, k) to Cfm(r, k). In particular, for f(z)

=

2: c(= wv i - k x (v-th Fourier coefficient of Tnf)

=

= wv1 - k11 1-k12 L dk-1c('1v/d2) d

where d runs over divisors of 17 modulo ox so that v/d 11"12

E

o*. This is

L d1-1c(w (r7v/d2)1-kc(17v/d2)) = 11"12 L d1 - k·( f , P(•; k, 17v/d 2)) = d

a ,d

1: 11k/2 L d i -kp( •; k, r,v/d 2)

.

d

Now T;,P(•; k, v)

= r,k12 L d 1-"P(•; k, vr,/d 2 ). d

From this, finally, ■ .

PROPOSITION The ring Hof operators on Cfm(r, k) generated by the Hecke operators is commutative. In particular, T~n = Tm 1'n for nt + n = o, and for p prime we have

¼Tp'" = Tpm+I + 1Vp ¼m-1. Proof This result is a consequence of the existence of the explicit upper triangular representatives for r \ Li(n), which is a consequence of the hypotheses on class number and units in o. The commutativity is a consequence of the explicit relations. Again, if n = 170 with t/ totally positive, there is a set of representatives X(n)

=

for r \ ~(n). It is a consequence of the Chinese Remainder Theorem that if m

+n=o

62

CHAPTER 1 Classical Theory of Hilbert Modular Forms ..

we have •

•

X(m)X(n)

= X(mn).

For the last assertion, let n be a totally positive generator of p; the Chine~ Remainder Theorem gives

Since n~ has the same action on modular for111s as does b, this gives the second explicit formula of the proposition. ■ •

Suppose that the ,veight is of the form k == (k 1 , ..• , k'") with each kj > 2 . Then there is an o rt hogonal l)asis for Cfm (l , k) consisting of simultaneous eigenvectors for all the Hecke operato rs. Let J be a to tally positive generator for the in verse different o*. For a simultaneous eigenvector

COROLLARY ...

f(z)

= L ~E

so that Ynf .

c(~)exp(2niTr~z)

o•

= ). (n)f, let '1

be a totally positive generat{!r for n; then

Proof The first assertion follows from the previous results by finite-dimensional linear algebra: A finite-dimensional Hilbert space with a commutative ring of self-adjoint operators has an.orthogonal basis consisting of simultaneous eigenvectors. We saw above that the effect of Tn on F·ourier coefficients is given by •

e-th Fourier coefficient of Tnf

= ,,l-lr./l L dlc-l c(~17/d 2 ) d

where d runs over divisors of 17 modulo ox so that ~/d E o*. Taking, o-th Fourier coefficient of Tnf

= {J gives

= 17 1- t12c(J77).

Therefore,

• Let/ E Cfm(r, k) be an eigenfunction for the Hecke operators norr11alized so that c(b) = 1, where 1J is a totally positive generator for o•. For a and /3 totally positive elen1ents of o so that o = ao + po,

COROLLARY

c(a/3t_~~y·"-equir.lt.1ri,1nt witt1 rcsi)ect tc• a o n t~.~(~irr1e11sj()l'lal ra qttotient 1.\l.J(2, f"') \ .S1J(2l l~\)./ [~ f{)f a suitabie ~>(:.m1,a. ..:t. (),pe11 subgrot~p fr~ of SL(2, /%, 01, anti Ct)11versely. \Ve \liew this a s asst~rting t!1at SJ.-4(2, F)\ S"/_.(2, t~ ) is a 14;-iivers{1l olJ,ject for these quotients. iii. Because tl1e 111atr1ces invol·ved i11 tr1e definitions of tl1e Hecke operato rs do no t often lie in S'L(2) :1nyy~;,ay, \1t1e rnight preler 10 be t:011sidering GL,(2, .P)\ GL(2, A); to inal ,111 auton1orpt1ic for1n on tl1is q t1oticrit is the iast ()bjc:,:t .i ve oft his secltn of classical