Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms

202 Pages·2011·1.03 MB·English

Michel Courtieu, Alexei Panchishkin
Non-Archimedean L-Functions
and Arithmetical Siegel Modular
Forms
SPINSpringer’s internal project number,if known
{ Monograph, LNM series {
August 3, 2003
Springer
Berlin Heidelberg NewYork
HongKong London
Milan Paris Tokyo

Preface
The present book is a renewed version of the LNM 1471 "Non-Archimedean
L-Functions of Hilbert and Siegel Modular Forms" by Alexei Panchishkin,
appearedin 1991.A partof this newbook usesthe resultsof the PhD Thesis
of Michel Courtieu (Grenoble, Institut Fourier, 2000).
The main subject of the book is the p-adic theoryof L-functions of Siegel
modularforms.Inthe caseofthe Riemannzetafunction,andofthe Dirichlet
L-functions, this theory goes back to the classical Kummer congruences for
Bernoulli numbers, and to their p-adic interpretation given by Kubota and
Leopoldt, and by Mazur. Using the technics of the p-adic integration, and
of the Rankin-Selberg convolution method, we construct a p-adic analytic
continuation of the standard L-functions of Siegel modular forms in a very
general case, that is, for any non-zero Satake p-parameter.
There are three basic new features of this second version, found recently
by Alexei Panchishkin:
1. The use of arithmeticalnearlyholomorphicSiegel modular forms,viewed
ascertainformalexpansionsofmanyvariableswith algebraiccoe(cid:14)cients.
2. Theuseofarithmeticaldi(cid:11)erentialoperatorsactingonnearlyholomorphic
Siegel modular forms.
3. The method of canonical projection allowing to deduce congruences be-
tweenspecialL-valuesfromcongruencesbetweenarithmeticalnearlyholo-
morphicSiegelmodularforms,viaasystematicuseofAtkin’sU -operator
p
(the Frobenius operator (cid:5)+(p) of A.N.Andrianov).
ThisnewmethodgivesaconceptualexplanationfortheformulasofManin
typeintheSiegelmodularcase,obtainedinLNM1471.Infact,thepreviously
used distributions admit a canonical lift to distributions with values in an
appropriate subspace of arithmetical Siegel modular forms. This lift depends
on a choice of a non-zero Satake parameter.
The book is intended to researches,to postgraduatestudents, and to pro-
fessors,interestedinrepresentationtheory,functionalanalysisandarithmetic
algebraic geometry. It contains, together with new results, much background
VI Preface
information about p-adic measures, their Mellin transforms, Siegel modular
forms, Hecke operators acting on them, Euler products etc.
It seems that the general methods developed in the book may have a
numberofotherapplications,inparticular,tofamiliesofSiegelmodularforms
and L-functions.
Saint-Hilaire du Touvet, August 2003
Contents
1 Non-Archimedean analytic functions::::::::::::::::::::::: 13
1.1 p-adic numbers and the Tate (cid:12)eld ......................... 13
1.2 Continuous and analytic functions ......................... 17
1.3 Distributions, measures, and the abstractKummer congruences 21
1.4 Iwasawaalgebra and the non-Archimedean Mellin transform .. 28
1.5 Admissible measures and their Mellin transform............. 35
1.6 Complex valued distributions, associated with Euler products . 38
2 Siegel modular forms and the holomorphic projection
operator ::::::::::::::::::::::::::::::::::::::::::::::::::: 45
2.1 Siegel modular forms and Hecke operators .................. 46
2.2 Theta series, Siegel-Eisenstein series and the Rankin zeta
function................................................ 62
2.3 Formulas for Fourier coe(cid:14)cients of the Siegel-Eisenstein series. 70
2.4 Holomorphic projection and Maass operator ................ 83
2.5 Explicit description of di(cid:11)erential operators................. 91
3 Arithmetical di(cid:11)erential operators::::::::::::::::::::::::: 95
3.1 Description of the Shimura di(cid:11)erential operators ............ 95
3.2 Nearly holomorphic Siegel modular forms................... 99
3.3 Algebraic di(cid:11)erential operatorsof Maass and Shimura........106
3.4 Arithmetical variables of nearly holomorphic forms ..........117
4 Admissible measures for standard L{functions:::::::::::::127
4.1 Congruences between modular forms and p-adic integration...127
4.2 Algebraic di(cid:11)erential operators and Siegel-Eisenstein
distributions ............................................133
4.3 A general result on admissible measures ....................144
4.4 The standard L-function .................................148
4.5 Algebraic linear forms on modular forms ...................165
4.6 Congruences and proof of the Main theorem ................174
VIII Contents
References:::::::::::::::::::::::::::::::::::::::::::::::::::::187
Index::::::::::::::::::::::::::::::::::::::::::::::::::::::::::195
Introduction
Kummer Congruences and Bernoulli Numbers
The starting point in the theory of zeta functions is the expansion of the
Riemann zeta-function (cid:16)(s) into the Euler product:
1
(cid:16)(s)= (1 p(cid:0)s)(cid:0)1 = n(cid:0)s (Re(s)>1):
(cid:0)
p n=1
Y X
Thesetofargumentssforwhich(cid:16)(s) isde(cid:12)nedcanbeextendedtoalls C,
2
s=1, and we may view C as the group of all continuous quasicharacters
6
C=Hom (R(cid:2);C(cid:2)); y ys
cont + 7(cid:0)!
of R(cid:2). The special values (cid:16)(1 k) at negative integers are rational numbers:
+ (cid:0)
B
k
(cid:16)(1 k)= ;
(cid:0) (cid:0) k
whereB areBernoullinumbers,whicharede(cid:12)nedbytheformalpowerseries
k
equality
1 B tn tet
eBt = n = ;
n! et 1
n=0 (cid:0)
X
and we know (by the Sylvester-Lipschitz theorem, see [Mi-St], [Kat3]) that
B
c Z implies ck(ck 1) k Z:
2 (cid:0) k 2
The theory of non-Archimedean zeta-functions originates in the work of
Kubota and Leopoldt [Ku-Le] containing p-adicinterpolationof these special
values. Their construction turns out to be equivalent to classical Kummer
congruences for the Bernoulli numbers, which we recall here in the following
form (see [Kat3]):
2 Introduction
Let p be a (cid:12)xed prime number, c>1 an integer prime to p. Put
(cid:16)(c)( k)=(1 pk)(1 ck+1)(cid:16)( k)
(p) (cid:0) (cid:0) (cid:0) (cid:0)
and let h(x) = n (cid:11) xi Z [x] be a polynomial over the ring Z of p-adic
i=0 i 2 p p
integers such that
P x Z = h(x) pmZ :
p p
2 ) 2
Then we have that
n
(cid:11) (cid:16)(c)( i) pmZ :
i (p) (cid:0) 2 p
i=0
X
This property expresses the fact that the numbers (cid:16)(c)( k) depend continu-
(p) (cid:0)
ously on k in the p-adic sense. This can be deduced from the known formula
for the sum of k-th powers:
N(cid:0)1
1
S (N)= nk = [B (N) B ]
k k+1 k+1
k+1 (cid:0)
n=1
X
in which B (x) =(x+B)k = k k B xk(cid:0)i denotes the Bernoulli polyno-
k i=0 i i
mial. Indeed, all summands in S (N) depend p-adic analytically on k, if we
k
P (cid:0) (cid:1)
restrict ourselves to numbers n, prime to p, so that the desired congruence
followsifweexpressthe numbers(cid:16)(c)( k)intermsof Bernoullinumbers(see
(p) (cid:0)
[Kat3]).
p{adic L{Functions and Mellin Transforms
Thedomainof de(cid:12)nitionof p-adiczeta functionsisthe p-adicanalyticLie
group
X =Hom (Z(cid:2);C(cid:2))
p cont p p
of all continuous p-adic characters of the pro(cid:12)nite group Z(cid:2), where C =Q
p p p
denotes the Tate (cid:12)eld (completion of an algebraic closure of the p-adic (cid:12)eld
Q ), so that all integers k can be viewed as the characters xk : y yk. Tbhe
p p 7!
construction of Kubota and Leopoldt is equivalent to the existence a p-adic
analyticfunction(cid:16) :X C withasinglepoleatthepointx=x(cid:0)1,which
p p ! p p
becomes a bounded holomorphic function on X after multiplication by the
p
elementary factor (x x 1), (x X ), and is uniquely determined by the
p p
(cid:0) 2
condition
(cid:16) (xk)=(1 pk)(cid:16)( k) (k 1):
p p (cid:0) (cid:0) (cid:21)
Thisresulthasaverynaturalinterpretationinframeworkofthetheoryof
non-Archimedeanintegration(duetoB.Mazur):thereexistsap-adicmeasure
(cid:22)(c) on Z(cid:2) with values in Z such that xk(cid:22)(c) = (cid:16)(c)( k). Indeed, if we
p p Z(cid:2)p p (p) (cid:0)
integrate h(x) over Z(cid:2) we exactly get the above congruence. On the other
p R
Introduction 3
hand,inordertode(cid:12)neameasure(cid:22)(c)satisfyingtheaboveconditionitsu(cid:14)ces
for any continuous function (cid:30) : Z(cid:2) Z to de(cid:12)ne its integral (cid:30)(x)(cid:22)(c).
p ! p Z(cid:2)p
Forthispurposeweapproximate(cid:30)(x)byapolynomial(forwhichtheintegral
R
is already de(cid:12)ned), and then pass to the limit.
The important feature of the construction is that it equally works for
primitive Dirichlet characters (cid:31) modulo a power of p. Throughout the book
we (cid:12)x an embedding
i :Q, C (0.1)
p p
!
and we shall identify Q with a sub(cid:12)eld of C and of C . Then a Dirichlet
p
character (cid:31):(Z=Z )(cid:2) Q(cid:2) becomes an element of the torsion subgroup
pN !
Xtors X =Hom (Z(cid:2);C(cid:2))
p (cid:26) p cont p p
andtheaboveequalityalsoholdsforthespecialvaluesL( k;(cid:31))oftheDirich-
(cid:0)
let L-series
1
L(s;(cid:31))= (cid:31)(n)n(cid:0)s = (1 (cid:31)(p)p(cid:0)s)(cid:0)1;
(cid:0)
n=1 p
X Y
so that we have
(cid:16) ((cid:31)xk)=i (1 (cid:31)(p)pk)L( k;(cid:31)) (k 1; k Z; (cid:31) Xtors): (0.2)
p p p (cid:0) (cid:0) (cid:21) 2 2 p
(cid:2) (cid:3)
The original construction of T. Kubota and H.W. Leopoldt [Ku-Le] was suc-
cessesfully used by K. Iwasawa [Iw] for the description of the class groups of
cyclotomic(cid:12)elds.Sincethen the classoffunctionsadmittingp-adicanalogues
has gradually extended.
Zeta-functions(ofcomplexvariable)canbeattachedascertainEulerprod-
uctstovariousobjectssuchasdiophantineequations,representationsofGalois
groups, modular forms etc..., and they play a crucial role in modern number
theory. Deep interrelations between these objects discovered in last decades
arebasedonidentitiesforthecorrespondingzetafunctionswhichpresumably
all(cid:12)tintoageneralconceptofLanglandsofL-functionsassociatedwithauto-
morphic representationsof a reductive group G overa number (cid:12)eld K. From
this pointof view the study of arithmeticpropertiesof thesezeta functions is
becoming especially important.
The theory of modular symbols (due to B. Mazur and Y.I. Manin, see
[Man1],[Man3],[Maz-SD])providedanon-Archimedeanconstructionoffunc-
tions, which correspond to the case of the group G=GL over K =Q. Sev-
2
eralauthors(includingP.Deligne,K.A.Ribet,N.M.Katz,P.F.Kur(cid:20)canovand
others, see [De-Ri], [Kat1], [Kat2], [Kur(cid:20)c1], [Kur(cid:20)c2], [Sho], [Vi1], [Vi2]) inves-
tigated this problem for the case G = GL and GL over totally real (cid:12)elds
1 2
and (cid:12)elds of CM-type (i.e. totally imaginary quadratic extentions of totally
real (cid:12)elds). But the case of more general reductive groups remained unclear
until the mid-eighties although important complex analytic properties of the
4 Introduction
LanglandsL-functionshadbeenproved.Inlastdecadesageneralapproachto
construction of non-Archimedean L-functions associated with various classes
of automorphicforms was developed, in particular,for the case of symplectic
groups of even degree over K = Q and the group G = GL GL over a
2 2
(cid:2)
totally real (cid:12)eld K.
ThemaintooloftheappearingtheoryisthesystematicuseoftheRankin-
Selbergmethodforobtainingbothcomplex-valuedandp-adicdistributionsas
certain integrals involving cusp forms and Eisenstein series. By this method
we constructed non-Archimedean analogues of the standard zeta functions
attached to Siegel cusp forms of even degree and of su(cid:14)ciently large weight.
p{adic L{Functions of Siegel Modular Forms.
For a Siegel modular form f(z) of degree m and weight k, which is an
eigenfunction of the Hecke algebra, and for each prime number p one can
de(cid:12)ne the Satake p-parameters of f denoted by (cid:11) (p) with i = 0;1; ;m.
i
(cid:1)(cid:1)(cid:1)
In this introduction we assume for simplicity that f is a modular form with
respect to the whole Siegel modular group (cid:0)m = Sp (Z), but in Chapter
m
3 we treat the greneral case of forms of level C with a Dirichlet character
modC. The standard zeta function of f is de(cid:12)ned by means of the Satake
p-parametersas the following Euler product:
(cid:31)(p) m (cid:31)(p)(cid:11) (p) (cid:31)(p)(cid:11) (p)(cid:0)1 (cid:0)1
i i
(s;f;(cid:31))= 1 1 1 ;
D p ((cid:18) (cid:0) ps (cid:19)i=1(cid:18) (cid:0) ps (cid:19)(cid:18) (cid:0) ps (cid:19))
Y Y
where(cid:31) isan arbitraryDirichlet character.Accordingto A.N.Andrianovand
V.L.Kalinin[An-Ka],thisfunctioncanberepresentedinaformofanintegral
convolution of f and a theta series with a Siegel-Eisenstein series as a ker-
nel. The construction of its p-adic analytic continuation is based on explicit
formulas for the special values of the standard zeta function, and this result
is equivalent to the existence of some generalized Kummer congruences for
these values. Before giving the precise statements of our results we introduce
the following normalized zeta functions
m
?(s;f;(cid:31))=(2(cid:25))(cid:0)m(s+k(cid:0)(m+1)=2)(cid:0)((s+(cid:14))=2) (cid:0)(s+k j) (s;f;(cid:31))
D 0 (cid:0) 1D
j=1
Y
(cid:0)(s;f;(cid:31))=(cid:0)((s+(cid:14))=2)(cid:0)1 ?(s;f;(cid:31)) @ A
D D
i(cid:14)(cid:25)1=2(cid:0)s
(cid:0)(s;f;(cid:31))= ?(s;f;(cid:31))
D (cid:0)((1 s+(cid:14))=2)D
(cid:0)
where (cid:14) =0 or 1 according as (cid:31)( 1)=1 or (cid:31)( 1)= 1, and let
(cid:0) (cid:0) (cid:0)
f(z)= a((cid:24))e ((cid:24)z) k
m 2Sm
(cid:24)>0
X