Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms: Second, Augmented Edition

191 Pages·2003·1.8 MB·English

by  Michel Courtieu

Introduction
Kummer Congruences and Bernoulli Numbers
The starting point in the theory of zeta functions is the expansion of the
Riemann zeta-function ζ(s) into the Euler product:
(cid:1) (cid:2)∞
ζ(s)= (1−p−s)−1 = n−s (Re(s)>1).
p n=1
The setofargumentss forwhichζ(s) isdefinedcanbe extendedto alls∈C,
s(cid:2)=1, and we may view C as the group of all continuous quasicharacters
C=Hom (R×,C×), y (cid:3)−→ys
cont +
of R×. The special values ζ(1−k) at negative integers are rational numbers:
+
B
ζ(1−k)=− k,
k
whereB areBernoullinumbers,whicharedefinedbytheformalpowerseries
k
equality
(cid:2)∞
B tn tet
eBt = n = ,
n! et−1
n=0
and we know (by the Sylvester-Lipschitz theorem, see [Mi-St], [Kat3]) that
B
c∈Z implies ck(ck−1) k ∈Z.
k
The theory of non-Archimedean zeta-functions originates in the work of
Kubota and Leopoldt [Ku-Le] containingp-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]):
M.CourtieuandA.Panchishkin:LNM1471,pp.1–12,2004.
(cid:1)c Springer-VerlagBerlinHeidelberg2004
2 Introduction
Let p be a fixed prime number, c>1 an integer prime to p. Put
ζ(c)(−k)=(1−pk)(1−ck+1)ζ(−k)
(p)
(cid:3)
and let h(x) = n α xi ∈Z [x] be a polynomial over the ring Z of p-adic
i=0 i p p
integers such that
x∈Z =⇒h(x)∈pmZ .
p p
Then we have that
(cid:2)n
α ζ(c)(−i)∈pmZ .
i (p) p
i=0
This property expresses the fact that the numbers ζ(c)(−k) depend continu-
(p)
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:2)−1
1
S (N)= nk = [B (N)−B ]
k k+1 k+1
k+1
n=1
(cid:3) (cid:4) (cid:5)
in which B (x) =(x+B)k = k k B xk−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
restrict ourselves to numbers n, prime to p, so that the desired congruence
followsifweexpressthe numbersζ(c)(−k)intermsofBernoullinumbers(see
(p)
[Kat3]).
p–adic L–Functions and Mellin Transforms
Thedomainofdefinitionofp-adiczeta functions isthe p-adicanalyticLie
group
X =Hom (Z×,C×)
p cont p p
(cid:6)
of all continuous p-adic characters of the profinite group Z×, where C =Q
p p p
denotes the Tate field (completion of an algebraic closure of the p-adic field
Q ), so that all integers k can be viewed as the characters xk : y (cid:3)→ yk. The
p p
construction of Kubota and Leopoldt is equivalent to the existence a p-adic
analyticfunctionζ :X →C withasinglepoleatthepointx=x−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
condition
ζ (xk)=(1−pk)ζ(−k) (k ≥1).
p p
Thisresulthasaverynaturalinterpretationinframeworkofthe theoryof
non-Archimedeanintegration(duetoB.Mazur):thereexistsap-adicmeasure
(cid:7)
µ(c) on Z× with values in Z such that xkµ(c) = ζ(c)(−k). Indeed, if we
p p Z×p p (p)
integrate h(x) over Z× we exactly get the above congruence. On the other
p
Introduction 3
hand,inordertodefineameasureµ(c)satisfyingtheabovecondit(cid:7)ionitsuffices
for any continuous function φ : Z× → Z to define its integral φ(x)µ(c).
p p Z×p
Forthispurposeweapproximateφ(x)byapolynomial(forwhichtheintegral
is already defined), and then pass to the limit.
The important feature of the construction is that it equally works for
primitive Dirichlet characters χ modulo a power of p. Throughout the book
we fix an embedding
i :Q(cid:6)→C (0.1)
p p
and we shall identify Q with a subfield of C and of C . Then a Dirichlet
p
character χ:(Z/ZpN)× →Q× becomes an element of the torsion subgroup
Xtors ⊂X =Hom (Z×,C×)
p p cont p p
andtheaboveequalityalsoholdsforthespecialvaluesL(−k,χ)oftheDirich-
let L-series
(cid:2)∞ (cid:1)
L(s,χ)= χ(n)n−s = (1−χ(p)p−s)−1,
n=1 p
so that we have
(cid:8) (cid:9)
ζ (χxk)=i (1−χ(p)pk)L(−k,χ) (k ≥1, k ∈Z, χ∈Xtors). (0.2)
p p p p
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
cyclotomicfields.Sincethenthe classoffunctions admittingp-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
allfitintoageneralconceptofLanglandsofL-functionsassociatedwithauto-
morphic representations of a reductive group G over a number field K. From
this pointofview the study ofarithmetic properties ofthese zeta 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.Kurˇcanovand
others, see [De-Ri], [Kat1], [Kat2], [Kurˇc1], [Kurˇc2], [Sho], [Vi1], [Vi2]) inves-
tigated this problem for the case G = GL and GL over totally real fields
1 2
and fields of CM-type (i.e. totally imaginary quadratic extentions of totally
real fields). 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 automorphic forms 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
totally real field 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 sufficiently 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
define the Satake p-parameters of f denoted by α (p) with i = 0,1,··· ,m.
i
In this introduction we assume for simplicity that f is a modular form with
respect to the whole Siegel modular group Γm = Sp (Z), but in Chapter
m
3 we treat the greneral case of forms of level C with a Dirichlet character
ψmodC. The standard zeta function of f is defined by means of the Satake
p-parameters as the following Euler product:
(cid:1)(cid:10)(cid:11) χ(p)(cid:12)(cid:1)m (cid:11) χ(p)α (p)(cid:12)(cid:11) χ(p)α (p)−1(cid:12)(cid:13)−1
D(s,f,χ)= 1− 1− i 1− i ,
ps ps ps
p i=1
where χ is an arbitraryDirichlet character.Accordingto A.N.Andrianov and
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
 
(cid:1)m
D(cid:1)(s,f,χ)=(2π)−m(s+k−(m+1)/2)Γ((s+δ)/2) Γ(s+k−j)D(s,f,χ)
j=1
D−(s,f,χ)=Γ((s+δ)/2)−1D(cid:1)(s,f,χ)
iδπ1/2−s
D−(s,f,χ)= D(cid:1)(s,f,χ)
Γ((1−s+δ)/2)
where δ =0 or 1 according as χ(−1)=1 or χ(−1)=−1, and let
(cid:2)
f(z)= a(ξ)e (ξz)∈Sk
m m
ξ>0
Introduction 5
be the Fourier expansion of the Siegel cusp form f(z) of weight k, the sum is
extended over all positive definite half integral m×m-matrices, z ∈H ,
m
H ={z ∈GL (C)|tz =z, Im(z) is positive definite}
m m
istheSiegelupperhalfplaneofdegreemande (z)=exp(tr(2πiz)).Assume
m
that k >2m+2 and that m is even.
Theorem A (Algebraic properties of the special values of the stan-
dard zeta functions).
(a) For all integers s with 1 ≤ s ≤ k−δ−m and Dirichlet characters χ
such that χ2 is non-trivial for s=1, we have that
(cid:9)f,f(cid:10)−1D+(s,f,χ)∈K =Q(f,Λ ,χ),
f
where K = Q(f,Λ ,χ) denotes the field generated by Fourier coefficients of
f
f, by the eigenvalues Λ (X) of Hecke operators X on f, and by the values of
f
the character χ.
(b) For all integers s with 1−k+δ+m≤s≤0 we have that
(cid:9)f,f(cid:10)−1D−(s,f,χ)∈K.
We may suppose without loss of generality that a(ξ) ∈ Q for all ξ > 0,
and let us assume also that a(ξ ) = 1 for some ξ > 0 with det(2ξ ) = 1.
0 0 0
Moreover, we make the essential assumption that a (ξ ) (cid:2)= 0 for the Fourier
0 0
coefficient a (ξ ) of the p-auxiliary form
0 0
(cid:2)
f (z)= a (ξ)e (ξz)
0 0 m
ξ>0
definedinthefirstsectionofchapter2bytheequality(2.57).Onthecontrary,
it is possible that α (p) is not a p-adic unit (see Theorem B below). We call
0
the case |α (p)| < 1 supersingular (i.e. non-p-ordinary). Note that in the
0 p
previous book ([Pa6], p 4), only the p-ordinary case was treated. According
to its definition, this form satisfies the following multiplicativity property
a (pNξ)=α (p)Na (ξ).
0 0 0
The importance of the above non-vanishing condition of a (ξ ) was pointed
0 0
out to the second author by S. Bo¨cherer.
Theorem B (non-Archimedean interpolation of the standard zeta
functions). Under the assumptions as above for each integer c>1 prime to
p there exist two C -analytic functions
p
Dc+(x,f), Dc−(x,f):X →C ,
p p
with the following properties:
6 Introduction
(i) for all pairs (s,χ) suchthat χ∈Xtors is a non-trivial Dirichlet charac-
p
ter, s∈Z with 1≤s≤k−δ−m, s≡δ (mod 2), and for s=1 the character
χ2 is non-trivial, the following equality holds
(cid:18) (cid:19)
G (χ)Cm(s+k−1−m) Cs D+(s,f,χ¯)
Dc+(χxs,f)=i m χ χ (1−χ¯2(c)c−2s)
p p α (C )2 G(χ¯) (cid:9)f,f(cid:10)
0 χ
(ii) for all pairs (s,χ) such that χ∈Xtors is a non-trivial Dirichlet char-
p
acter and s ∈ Z with 1−k+δ +m ≤ s ≤ 0, s (cid:2)≡ δ (mod 2), the following
equality holds
(cid:18) (cid:19)
G (χ)Cm(s+k−1−m) D−(s,f,χ¯)
Dc−(χxs,f)=i m χ (1−χ2(c)c2s−2)
p p α (C )2 (cid:9)f,f(cid:10)
0 χ
where (cid:2)
G (χ)= χ(det(h))e (h/C )
m m χ
h∈Mm(Z)modCχ
denotes theGauss sumof degree m of theprimitive Dirichlet character χmod
C , C =pNχ, α (C )=α (p)Nχ, G(χ)=G (χ).
χ χ 0 χ 0 1
(iii) if ord α (p) = 0 then the holomorphic functions in (i), (ii) are
p 0
bounded C -analytic functions;
p
(iv) in the general case (but assuming that α (p) (cid:2)= 0) the holomorphic
0
functions in (i), (ii) belong to the type o(log(xh)) with h = [2ord α (p)]+1
p p 0
and they can berepresentedas theMellin transform of h-admissible measures;
(v) if h ≤ k−m then the functions Dc± are uniquely determined by the
above conditions (i) and (ii).
(see Main Theorem in §4.1.5 and Theorem 4.23).
Distributions with Values in Siegel Modular Forms.
One of the purposes of this book is to give a new conceptual construction
of admissible measures (in the sense of Amice-V´elu) attached to a standard
L-function of a Siegel cusp eigenform. For this purpose we use the theory of
p-adic integration with values in spaces of arithmetical nearly holomorphic
Siegel modular forms (in the sense of Shimura, see [Sh9]) defined over an O-
algebra A where O is the ring of integers in a finite extension K of Q . Often
p
we simply assume that A=C .
p
Arithmetical nearly holomorphic Siegel modular formsadmittwodifferentde-
scriptions: (cid:3)
• as formal power series g = a(ξ,R )qξ ∈ Q[[qBm]][R ] such that
i,j i,j
ξ∈Bm
for all R = (Ri,j)i,j=1,···,m = (4πIm(z))−1 = (2πi(z¯−z))−1 and z ∈ Hm in
the Siegel upper half plane of degree m the series converges to a C∞-Siegel
modular form of a given weight k and character ψ;
Introduction 7
•ascertainC∞-SiegelmodularformsgtakingvaluesinQatallCM-points
(up to a factor independnt of a concrete form).
In fact, we need to fix throughout the book two embeddings
i∞ :Q→C, ip :Q→Cp,
andweshalloftenviewthefieldQasasubfieldofbothCandC (thecomplex
p
andthe p–adic numbers)via these embeddings, omitting the symbolsi∞ and
(cid:6)
i . Here C =Q is the Tate field (the completion ofa fixed algebraicclosure
p p p
Q of Q ), which is endowed with a unique norm |·| such that |p| =p−1.
p p p p
We describe the action of certain arithmetical differential operators on
algebraically defined nearly holomorphic Siegel modular forms (in the sense
of Shimura, see [Sh9] over anO-algebraA where O is the ring of integers in a
finite extension K of Q . Often we simply assume that A=C .
p p
Theimportantpropertyofthesearithmeticaldifferentialoperatorsistheir
commutation with the Hecke operators (under an appropriate normalization,
see Theorem 3.13). The reason of these nice arithmetical properties is that
these(non-holomorphic)operatorscouldbealgebraicallyinterpretedinterms
oftheGauss-Manin connectionactingonthedeRhamcohomologysheavesof
the arithmetical compactification of Siegel modular varieties (see [Fa-Ch90],
[Ha2], [Ga-Ha]).
Denote by M = M(A) the A-module (or simply the C -vector space) of
p
nearly holomorphic Siegel modular forms . We show in Chapter 4 that the
actionofthese arithmeticaldifferentialoperatorsproducesnaturalfamilies of
distributions on a profinite group Y =limY with values in these A-modules
←− i
of nearly holomorphic Siegel modular forms inside a formal q-expansion ring
like A[[qB]][Ri,j] where B = Bm = {ξ = tξ ∈ Mm(Q)|ξ≥0, ξ half-integral} is
the semi-group, important for the theory of Siegel modular forms), and the
nearly holomorphic parameters (R ) = R correspond to the matrix R =
i,j
(4πIm(z))−1 in the Siegel modular case.
Weapplytothesedistributionsthecanonical projectorπ :M→Mαonto
α
the primary(characteristic)A-submodule associatedto anon-zeroeigenvalue
α ∈ A× of the Frobenius operator U = Π+(p) = U (the Atkin-Lehner oper-
p
ator in the Siegel modular case; Mα is the maximal A-submodule on which
U −αI is nilpotent). This operator acts similar to the trace operator lower-
ing the level of modular forms. On the other hand, U is invertible on Mα if
α∈A× so that one can glue its action on forms of various levels. In this way
one obtains the desired distributions with values in a finite dimensional vec-
torspacestartingfromnaturallydefineddistributionswithvaluesinspacesof
modular forms (like Siegel-Eisenstein distributions, theta distributions etc.).
In order to obtain from them numerically valued distributions interpolating
critical values attached to standard L-functions of Siegel modular forms one
appliesasuitablelinearformcomingfromthePeterssonscalarproduct(using
the Andrianov identity of Rankin type for the standard L-function).
8 Introduction
In previous works [Pa7], [Pa6] a non–Archimedean interpolation of these
special values was constructed fixing a non zero Satake p–parameterα (p) of
0
the cusp eigenform f. It follows that the normalized critical values D(s,f,χ)
of the standard L-function can be explicitely rewritten in terms of certain
C –valued integrals of admissible p–adic measures (over a profinite group of
p
the type Z× of p–adic units) provided the character χ is non–trivial. More
p
explicit description of these special values was given by M.Courtieu in his
PhD thesis ([Cour], Institut Fourier, 2000)using detailed study of the action
of the Shimura differential operator on Siegel modular forms.
In the present book we give a conceptual explanation of the p–adic prop-
erties satisfyied by the special values of the standard L-function D(s,f,χ),
where f is a Siegel cusp form of an even degree m and of weight k >2m+2,
χ is a varying Dirichlet character. We show that these admissible measures
canbeliftedtoarithmeticalnearlyholomorphicSiegelmodularformsstudied
by G.Shimura [Sh9]. This lifting is given by a universal sequence Φ±(χ) of
s
distributions with values in arithmetical nearly holomorphic Siegel modular
forms (for critical pairs (s,χ), see Proposition 4.21). It would be interesting
to extend these lifting results to Siegel cusp eigenforms of odd degree, using
the method of Bo¨cherer-Schmidt[Bo¨Sch].
Families of Siegel Modular Forms.
Note that the problem of construction of families of modular forms is
closely related to the context of Wiles’ proof [Wi] which is based on a Galois
cohomologicalconstructionofp-adicfamiliesofclassicalelliptcmodularforms.
It seems that a natural thing would be to try to extend constructions of such
familiestootherclassesofmodularforms;thepresentbookgivesanexample:
a canonicallift of previouslyknownp–adic distributions 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, and it produces
families by integration of arithmetical characters. A construction of rigid–
analytic families of modular distributions is given recently in [PaInv].
Applications of this theory to constructions of certain new p-adic families
of modular forms (families of Siegel-Eisenstein series, families of theta-series
with spherical polynomials...) is based on the following main sources:
• Serre’s theory of p-adic modular forms as certain formal q-expansions,
[Se2].
• Shimura’s theory of arithmeticity for nearly holomorphic forms, [Sh9].
• Hida’s theory of p-adic modular forms and p-adic Hecke algebras, [Hi5].
• Constructions of p-adic Siegel-Eisenstein series, and of p-adic Klingen–
Eisenstein series by the author, [Pa8].
Note that the eigenspaces M(α) of U are contained in the primary sub-
spaces Mα, and they where used by D. Kazhdan, B. Mazur, C.-G. Schmidt,
Introduction 9
see [KMS2000], in the p-ordinary case via a p–adic limit procedure. Notice
that we do not need a p–adic limit procedure, and we treat the general case
of any positive slope.
Remarks on modular forms of positive slope
Accordingto R.Coleman,F. GouvˆeaandB.Mazur,the structure ofmod-
ular forms of a given positive slope is more complicated than in the ordinary
case, even for elliptic modular forms (see the theory of ”ferns” in [Gou-Ma],
and [Co-Ma]). Our results mean that the p-adic behaviour of the standard
p-adic L-functions depends not only on the slope of a Siegel modular form
but also the eigenvalue itself, see also [PaInv].
This would be important for constructions of rigid–analytic families of
Colemantype [CoPB]intheSiegelmodularcase,andthe correspondingfam-
ilies of p-adic L-functions. Notice that the structure of weights is more com-
plicated in the Siegel modular case due to vector-valued modular forms.
It seems that besides fixing just a positive slope one needs some addi-
tionalandmore subtle parameters,givenprobablyby ananalogueof”ferns”,
which could provide a good understanding of overconvergency in the Siegel
modular case. In the future, it would be interesting to combine our approach
with geometric methods of Faltings–Chai [Fa-Ch90] and of Coleman–Mazur
[Co-Ma].
For a fixed positive integer N ∈N consider the profinite group
Y =limY , where Y =(Z/NpvZ)×.
← v v
v
There is a natural projection y :Y →Z×.
p p
Definition 0.1.a) For h∈N,h≥1 let Ch(Y,A) denote the A-module of
locally polynomial functions of degree <h of the variable y :Y →Z× (cid:6)→A×;
p p
in particular,
C1(Y,A)⊂Cloc−const(Y,A)
(theA-submoduleoflocallyconstantfunctions).WeadoptthenotationΦ(U):=
Φ(χU) for the characteristic function χU of an open subset U ⊂ Y. Let also
denote Cloc−an(Y,A) the A-module of locally analytic functions and C(Y,A)
the A-module of continuous functions so that
C1(Y,A)⊂Ch(Y,A)⊂Cloc−an(Y,A)⊂C(Y,A).
b) For a given positive integer h an h-admissible measure on Y with values in
M is an A-module homomorphism
Φ˜:Ch(Y,A)→M
such that for all a∈Y and for v →∞
10 Introduction
(cid:20) (cid:20)
(cid:20)(cid:21) (cid:20)
(cid:20) (cid:20)
(cid:20) (y −a )jdΦ˜(cid:20) =o(p−v(j−h)) for all j =0,1,··· ,h−1,
(cid:20) p p (cid:20)
a+(Npv)
p,M
where a =y (a).
p p
We adopt the notation (a) = a+(Npv) for both an element of Y and
v v
the corresponding open compact subset of Y.
U(p)–Operator and Method of Canonical projection.
We explain in Section 4.3 of Chapter 4, how to construct an h-admissible
measure Φ(cid:22)α :Ch(Y,A)→M(A) out of a sequence of distributions
Φ :C1(Y,A)→M
j
withvaluesinanA-module M =M(A)ofnearlyholomorphicmodularforms
over A (for all j ∈ N with j ≤ h−1, where A is an O-algebra, and α ∈ A×
is a fixed non-zero eigenvalue of the Frobenius operator U = U(p) = Π+(p)
acting on (cid:2)
g = a(ξ,n)Rnqξ ∈A[[qBm]][R ],
i,j
ξ∈Bm,n≥0
by (cid:2)
g|U(p)= a(pξ,n)(pR)nqξ
ξ∈Bm,n≥0
(overthecomplexnumbersR=(4π)−1Im(z)−1 andthisnotationcorresponds
to (cid:11) (cid:12)
(cid:2)
1 u
f|U(p)(z)= f| m =
k 0 p1
m m
tu=u∈Mm(Z)modp
(cid:2)
p−κm f((z+u)/p),
tu=u∈Mm(Z)modp
Im((z+u)/p)=Im(z/p)=Im(z)/p=(4π)−1(pR)−1
(we usethe Petersson-Andrianovnotationforthe actionofmatrices,Chapter
2). Then we consider (for an α∈ A×) the canonical projection operator π :
α
M(A)→M(A)α. We define an A-linear map
Φ(cid:22)α :Ch(Y,A)→M
on local monomials yj by
p
(cid:21)
yjdΦ(cid:22)α =π (Φ ((a) ))
p α j v
(a)v
where Φ :C1(Y,A)→M(A) are certain M(A)-valued distributions on Y for
j
all j =0,1,...,h−1.