L-Functions of Siegel Modular Forms, Their Families
and Lifting Conjectures
Alexei PANCHISHKIN
Institut Fourier, Université Grenoble-1
B.P.74, 38402 St.Martin d'Hères, FRANCE
A talk for the Conference "Modulformen"
October 29 to November 2, 2007 (Oberwolfach, Germany)
October 2007
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
1 / 27
Introduction
1)
L-
functions of Siegel modular forms
2) Motivic
L-functions
3) Critical values and
4) A lifting from
for
Spg , and their analytic properties
p -adic
constructions for the spinor
GSp 2m × GSp 2m
5) Constructions of
p -adic
to
GSp 4m
(of genus
L-functions
g = 4m )
families of Siegel modular forms
Γ = Spg (Z) ⊂ SL2g (Z) be the Siegel modular group of genus g .
T(p ) = T(1, · · · , 1, p , · · · , p ) the Hecke p -operator, and
| {z } | {z }
g
g
[p]g = p I2g = T(p , · · · , p ) the scalar Hecke operator for Spg .
| {z }
2g
Let
Let
p
be a prime,
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
2 / 27
The Fourier expansion of a Siegel modular form.
Let
f =
genus
n
X
T∈Bn
a(T)q T ∈ Mnk
be a Siegel modular form of weight
on the Siegel upper-half plane
The formal
Fourier expansion
of
f
Hn = {z ∈ Mn (C) |
k
and of
Im(z ) > 0}.
uses the symbol
q T = exp(2π i tr(T z )) =
n
Y
i =1
qiiTii
Y
i <j
2Tij
qij
√
∈ C[[q11 , . . . , qnn ]][qij , qij−1 ]i ,j =1,··· ,m , where qij = exp(2π( −1zi ,j )),
t
is in the semi-group Bn = {T = T ≥ 0|T half-integral}.
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
and
T
3 / 27
Satake parameters of an eigenfunction of Hecke operators
f ∈ Mnk
Hecke operators f 7−→ f |T ,
T ∈ Ln,p for all primes p , hence f |T = λf (T )f .
Then all the numbers λf (T ) ∈ C dene a homomorphism λf : Ln,p −→ C
× n +1
given by a (n + 1)-tuple of complex numbers (α0 , α1 , · · · , αn ) = (C )
(the Satake parameters of f ): λf (T ) = Satake(T ) for
x0 := α0 , . . . , xn := αn , where Satake : Lp,n → Q[x0± , x1± , . . . , xn± ]Wn is the
Suppose that
an eigenfunction of all
Satake isomorphism
Recall that
Alexei PANCHISHKIN (Grenoble)
α02 α1 · · · αn = p kn−n(n+1)/2 .
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
4 / 27
L-functions, functional equation and motives for Spn (see
[Pa94], [Yosh01])
One denes
n
Y
Y
(1 − α0 αi1 · · · αir X ),
r =1 1≤i1 <···<ir ≤n
n
Y
Rf ,p (X ) = (1 − X ) (1 − αi−1 X )(1 − αi X ) ∈ Q[α0±1 , · · · , αn±1 ][X ].
i =1
Then the spinor L-function L(Sp (f ), s ) and the standard L-function
L(St (f ), s , χ) of f (for s ∈ C, and for all Dirichlet characters) χ are
Qf ,p (X ) = (1 − α0 X )
dened as the Euler products
L(Sp (f ), s , χ) =
L(St (f ), s , χ) =
Y
p
Y
Alexei PANCHISHKIN (Grenoble)
p
Qf ,p (χ(p )p −s )−1
Rf ,p (χ(p )p −s )−1
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
5 / 27
Relations with
L-functions and motives for Spn
Following [Pa94] and [Yosh01], these functions are conjectured to be motivic
for all
k > n:
L(Sp (f ), s , χ) = L(M (Sp (f ))(χ), s ), L(St (f ), s ) = L(M (St (f ))(χ), s ),
and the motives
M (Sp (f ))
and
M (St (f ))
are
pure
if
f
where
is a genuine cusp form
(not coming from a lifting of a smaller genus):
M (Sp (f ))
of weight
Q with coecients in Q(λf (n))n∈N of rank 2n ,
w = kn − n(n + 1)/2, and of Hodge type ⊕p,q H p,q , with
is a motive over
p = (k − i1 ) + (k − i2 ) + · · · + (k − ir ),
q = (k − j1 ) + (k − j2 ) + · · · + (k − is ),
1
(3.1)
where
r + s = n,
≤ i1 < i2 < · · · < ir ≤ n, 1 ≤ j1 < j2 < · · · < js ≤ n,
{i1 , · · · , ir } ∪ {j1 , · · · , is } = {1, 2, · · · , n};
M (St (f )) is a motive over Q with coecients
2n + 1, of weight w = 0, and of Hodge type
H 0,0 ⊕ni=1 (H −k +i ,k −i ⊕ H k −i ,−k +i ).
Alexei PANCHISHKIN (Grenoble)
L-Functions of
in
Siegel Modular Forms
Q(λf (n))n∈N
of rank
Oberwolfach 2007
6 / 27
A functional equation
Following general Deligne's conjecture [De79] on the motivic
L-function
L-functions,
the
satisfy a functional equation determined by the Hodge structure of a
motive:
Λ(Sp (f ), kn − n(n + 1)/2 + 1 − s ) = ε(f )Λ(Sp (f ), s ),
k 2n−2
Λ(Sp (f ), s ) = Γn,k (s )L(Sp (f ), s ), ε(f ) = (−1)
where
,
−s s ), Γ (s ) = Γ (s )Γ (s − k + 2), and
Γ1,k (s ) = Γ
C
C
2,k
QC (s ) = 2(2π) hΓ(
Γn,k (s ) = p<q ΓC (s − p ) p,q ΓR (s − (w /2))a+ ΓR (s + 1 − (w /2))a− for
non-negative integers a+ and a− , with a+ + a− = hw /2,w /2 , and
ΓR (s ) = π −s /2 Γ(s /2), where hp,q = dimC H p,q .
In particular, for n = 3 and k ≥ 5, Λ(Sp (f ), s ) = Λ(Sp (f ), 3k − 5 − s ),
some
Λ(Sp (f ), s ) = ΓC (s )ΓC (s − k + 3)ΓC (s − k + 2)ΓC (s − k + 1)L(Sp (f ), s ).
For
k ≥5
the critical values in the sense of Deligne [De79] are :
s = k , · · · , 2k − 5.
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
7 / 27
A study of the analytic properties of
L(Sp(f ), s )
(compare with [Vo]).
One could try to use a link between the eigenvalues
coecients
T ∈ Bn
af (T),
where
T ∈ D(Γ, S )
λf (T )
and the Fourier
runs through the Hecke operators, and
runs over half-integral symmetric matrices.
D (X ) =
∞
X
T(p δ )X δ =
δ=0
E (X )
,
F (X )
where
E (X )
= 1 − p 2 T2 (p 2 ) + (p 2 − p + 1)(p 2 + p + 1)[p]3 X 2 + (p + 1)p 4 T(p )[p]3 X 3
− p 7 [p]3 T2 (p 2 ) + (p 2 − p + 1)(p 2 + p + 1)[p]3 X 4 + p 15 [p]33 X 6 ∈ LZ [X ].
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
8 / 27
Computing a formal Dirichlet series
Knowing
E (X ),
DE (s ) =
one computes the following formal Dirichlet series
∞
X
TE (h)h−s =
Y
p
DE ,p (p −s ),
where
h =1
∞
X
D (X )
1
DE ,p (X ) =
TE (p δ )X δ = p
=
∈ D(Γ, S )[[X ]].
E (X )
F (X )
δ=0
Hence,
L(Sp (f ), s ) =
P∞
−s
h=1 λf (TE )(h)h .
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
9 / 27
Main identity
For all
T
one obtains the following identity
af (T)L(Sp (f ), s ) =
f |TE (h) =
X
T∈Bn
∞
X
af (T, E , h)h−s ,
where
h =1
af (T, E , h)q T .
Such an identity is an unavoidable step in the problem of analytic
continuation of
L(Sp (f ), s )
and in the study of its arithmetical implications:
one obtains a mean of computing special values out of Fourier coecients.
We hope also to use the Jacobi forms for the study of
L(Sp (f ), s ).
For the standard
L(St (f ), s ),
Alexei PANCHISHKIN (Grenoble)
L-functions
(see [Pa94], [CourPa], and [Boe-Schm]).
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
10 / 27
Critical values, periods and
p-adic L-functions for Sp3
A general conjecture by Coates, Perrin-Riou (see [Co-PeRi], [Co], [Pa94]), predicts
that for
n=3
and
k > 5,
the motivic function
L(Sp (f ), s ),
admits a
p -adic
analogue.
The known conjectural condition for the existence of bounded
p -adic
ordp (α0(p)) = 0.
for a motive M of rank d ,
Newton p-Polygone at (d /2)=Hodge Polygone at (d /2)
h
Otherwise, one obtains p -adic L-functions of logarithmic growth o (log (·))
with h = [2ordp (α0 (p ))] + 1, 2ordp (α0 (p )) =the dierence
L-functions
in this case takes the form
Recall : this condition says that
Newton
p -Polygone
at (d /2)Hodge Polygone at (d /2)
For the unitary groups, this condition for the existence of
L-functions
p -adic
was discussed in [Ha-Li-Sk].
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
11 / 27
n
Motive of the Rankin product of genus = 2
Let f and g be two Siegel cusp eigenforms of weights k and l , k > l , and let
M (Sp (f )) and M (Sp (g )) be the spinor motives of f and g . Then M (Sp (f )) is
a motive over Q with coecients in Q(λf (n ))n∈N of rank 4, of weight
w = 2k − 3, and of Hodge type H 0,2k −3 ⊕ H k −2,k −1 ⊕ H k −1,k −2 ⊕ H 2k −3,0 , and
M (Sp (g )) is a motive over Q with coecients in Q(λg (n))n∈N of rank 4, of
0,2l −3 ⊕ H l −2,l −1 ⊕ H l −1,l −2 ⊕ H 2l −3,0 .
weight w = 2l − 3, and of Hodge type H
The tensor product M (Sp (f )) ⊗ M (Sp (g )) is a motive over Q with coecients
in Q(λf (n ), λg (n ))n∈N of rank 16, of weight w = 2k + 2l − 6, and of Hodge
type
H 0,2k +2l −6 ⊕ H l −2,2k +l −4 ⊕ H l −1,2k +l −5 ⊕ H 2l −3,2k −3
H k −2,k +2l −4 ⊕ H k +l −4,k +l −2 ⊕ H k +l −3,k +l −3 ⊕ H k +2l −5,k −1
+
H k −1,k +2l −5 ⊕ H k +l −3,k +l −3 ⊕ H k +l −2,k +l −4 ⊕ H k +2l −4,k −2
H
2k −3,2l −3
−
⊕H
Alexei PANCHISHKIN (Grenoble)
2k +l −5,l −1
⊕ H 2k +l −4,l −2 ⊕ H 2k +2l −6,0 .
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
12 / 27
Motivic
L-functions:
analytic properties
L-functions, applied for a Siegel cusp
F for the Siegel modular group Sp4 (Z) of genus n = 4 and of weight k > 5,
Λ(Sp (F ), s ) = Λ(Sp (F ), 4k − 9 − s ), where
Following Deligne's conjecture [De79] on motivic
eigenform
one has
Λ(Sp (F ), s ) = ΓC (s )ΓC (s − k + 4)ΓC (s − k + 3)ΓC (s − k + 2)ΓC (s − k + 1)
× ΓC (s − 2k + 7)ΓC (s − 2k + 6)ΓC (s − 2k + 5)L(Sp (F ), s ),
(compare this functional equation with that given in [An74], p.115).
m = 2 and for two cusp eigenforms f and g for Sp2 (Z) of
k , l , k > l + 1, Λ(Sp (f ) ⊗ Sp (g ), s ) = ε(f , g )Λ(Sp (f ) ⊗ Sp (g ), 2k + 2l − 5 − s ),
|ε(f , g )| = 1, where
On the other hand, for
weights
Λ(Sp (f ) ⊗ Sp (g ), s ) = ΓC (s )ΓC (s − l + 2)ΓC (s − l + 1)ΓC (s − k + 2)
× ΓC (s − k + 1)ΓC (s − 2l + 3)ΓC (s − k − l + 2)ΓC (s − k − l + 3)
× L(Sp (f ) ⊗ Sp (g ), s ).
ΓC (s ) = ΓR (s )ΓR (s + 1).
a+ = a− = 1 in this case, and the conjectural motive
M (Sp (f )) ⊗ M (Sp (g )) does not admit critical values.
We used here the Gauss duplication formula
Notice that
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
13 / 27
A holomorphic lifting from
GSp2m × GSp2m to GSp4m :
a
conjecture
Conjecture (on a lifting from
GSp 2m × GSp 2m to GSp 4m )
Let f and g be two Siegel modular forms of genus 2m and of weights k > 2m and
l = k − 2m. Then there exists a Siegel modular form F of genus 4m and of weight k with the
Satake parameters γ0 = α0 β0 , γ1 = α1 , γ2 = α2 , · · · , γ2m = α2m , γ2m+1 = β1 , · · · , γ4m = β2m
for suitable choices α0 , α1 , · · · , α2m and β0 , β1 , · · · , β2m of Satake's parameters of f and g .
One readily checks that the Hodge types of M (Sp (f )) ⊗ M (Sp (g )) and M (Sp (F )) are the
same (of rank 24m ) (it follows from the above description (3.1), and from Künneth's-type
formulas).
An evidence for this version of the conjecture comes from Ikeda-Miyawaki
constructions ([Ike01], [Ike06], [Mur02]): let k be an even positive integer,
h ∈ S2k (Γ1 ) a normalized Hecke eigenform of weight 2k ,
g := F2n (h) ∈ Sk +n (Γ2n ) the IkedaDuke-Imamoglu lift of h of genus 2n
(we assume k ≡ n mod 2, n ∈ N).
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
14 / 27
f ∈ Sk +n+r (Γr ) be an arbitrary Siegel cusp eigenform of genus r and
k + n + r , with n, r ≥ 1.
Let us consider the lift F2n+2r (h ) ∈ Sk +n+r (Γ2n+2r ), and the integral
Z
z
k +n +r −1 0
F h ,f =
F2n+2r (h)
f (z 0 )(det Im(z0 )
dz ∈ Sk +n+r (Γ2n+r )
0
z
Γr \Hr
Next let
weight
which denes the
IkedaMiyawaki lift,
Under the non-vanishing condition on
where
F h ,f ,
L(s , Fh,f , St ) = L(s , f , St )
If we take
2n
Y
i =1
z ∈ H2n+r , z 0 ∈ Hr .
one has the equality
L(s + k + n + r − i , h)
n = m, r = 2m, k + n + r := k + 3m,
then an example of the
validity of this version of the conjecture is given by
(f , g ) = (f , F2m (h)) 7→ Fh,f ∈ Sk +3m (Γ4m ),
(f , g ) = (f , F2m (h)) ∈ Sk +3m (Γ2m ) × Sk +m (Γ2m ).
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
15 / 27
Another evidence comes from Siegel-Eiseinstein series
f = Ek2m
of even genus 2m and weights
k
and
and
m
g = Ek2−
2m
k − 2m :
we have then
α0 = 1, α1 = p k −2m , · · · , α2m = p k −1 ,
β0 = 1, β1 = p k −4m , · · · , β2m = p k −2m−1 ,
then we have that
γ0 = 1, γ1 = p k −4m , · · · , γ2m = p k −1 ,
are the Satake parameters of the Siegel-Eisenstein series
F = Ek4m .
Also, the compatibility of the conjecture with the formation of
Klingen-Eisenstein series was checked after a discussion with
Prof. Tetsushi ITO (Kyoto University) in Luminy in July 2007.
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
16 / 27
Remark
If we compare the L-function of the conjecture (given by the Satake parameters
γ0 = α0 β0 , γ1 = α1 , γ2 = α2 , · · · , γ2m = α2m , γ2m+1 = β1 , · · · , γ4m = β2m for suitable
choices α0 , α1 , · · · , α2m and β0 , β1 , · · · , β2m of Satake's parameters of f and g ), we see
that it corresponds to the tensor product of spinor L-functions, and is not of the same
type as that of the Yoshida's lifting [Yosh81], which is a certain product of Hecke's
L-functions.
We would like to mention in this context Langlands's functoriality: The denominators of
L-series belong to local Langlands L-factors
L-groups). If we consider the homomorphisms
our
L GSp
2m
(attached to representations of
= GSpin(4m + 1) → GL22m , L GSp4m = GSpin(8m + 1) → GL24m ,
we see that our conjecture is compatible with the homomorphism of
L-groups
GL22m × GL22m → GL24m , (g1 , g2 ) 7→ g1 ⊗ g2 , GLn (C) = L GLn .
However, it is unclear to us if Langlands's functoriality predicts a
holomorphic Siegel modular form as a lift.
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
17 / 27
Constructing
p-adic L-functions
s it is possible to use certain p -adic
L-functions. We can use such parameters
Together with complex parameter
parameters in order to study the
as the twist with Dirichlet character on one hand, and the weight
parameter in the theory of families of modular forms on the other.
L-functions
L(s , π ⊗ χ, r ) related to an automorphic representation π of an algebraic
group G over a number eld, and the twists with Dirichlet character χ.
In several cases one can compute the values of the automorphic
A usual tool is to represent these special values as integrals giving both
p -adic-analytic continuation.
In this way one can treat the cases G = GL2 × GL2 × GL2 ,
G = GL2 × GSp2m , and probably G = GSp2m × GSp2m using the
doubling method and its p -adic versions. Computations involve dierential
complex-analytic and
operators acting on modular forms.
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
18 / 27
A
p-adic approach
for prime number p . Let us x the
ip
embedding Q ,→ Cp and consider the algebraic numbers as numbers p -adic
P∞
n
over ip . For p -adic family k 7→ fk =
n=1 an (k )q ∈ Q[[q ]] ⊂ Cp [[q ]]
Fourier coecients an (k ) of fk and one of Satake's p -parameters
(1)
α(k ) := αp (k ) are given by the certain p -adic analytic functions
k 7→ an (k ) for (n, p ) = 1. A typical example of a p -adic family is given by
Consider Tate's eld
b
Cp = Q
p
the Eisenstein series.
an (k ) =
X
d |n,(d ,p)=1
d k −1 , fk = Ek , αp (k ) = 1, αp (k ) = p k −1 .
(1)
p -adic
(2)
σ>0
(1)
the slope σ = ordp (αp (k )) (and ask
to be constant in a p -adic neighborhood of weight k ). An example for
p = 7, f = ∆, k = 12, a7 = τ (7) = −7 · 2392, σ = 1 is given by
The existence of the
family of cusp forms of positive slope
was shown by Coleman. We dene
it
R. Coleman in [CoPB].
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
19 / 27
Motivation for considering of
p-adic families
comes from Birch and Swinnerton-Dyer's conjecture, see [Colm03]. For a
E by Wiles
f . One can try to approach
k = 2, s = 1 from the direction, taking k → 2, instead of s → 1, this leads
to a formula linking the derivative over s at s = 1 of the p -adic L-function
with the derivative over k at k = 2 of the p -adic analytic function αp (k ),
d αp (k ) 0
see in [CST98]: Lp ,f (1) = Lp (f )Lp ,f (1) with Lp (f ) = −2
k =2 .
dk
cusp eigenform
f = f2 ,
corresponding to an elliptic curve
[Wi95], we consider a family containing
The validity of this formula needs the existence of our two variable
L-function!
In order to construct
p -adic
p -adic L-function of two variables (k , s ), the theory
H -admissible measures with integer H
of
integration is used. The
related to σ , which appear in this construction, are obtained from
H -admissible measures with values in various rings of modular forms,
in
particular nearly holomorphic modular forms.
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
20 / 27
Nearly holomorphic modular forms and the method of
canonical projection
Let
A
be a commutative eld. There are several
studying special values of
L-functions
p -adic
approaches for
using the method of canonical
projection (see [PaTV]). In this method the special values and modular
symbols are considered as
with coecients in
A.
A-linear
forms over spaces of modular forms
Nearly holomorphic ([ShiAr]) modular forms are
certain formal series
g=
∞
X
n =0
a(n; R )q n ∈ A[[q ]][R ]
A = C, z = x + iy ∈ H, R = (4π y )−1 , the series
∞
converge to a C -modular form over H of given weight k and Dirichlet
character ψ . The coecients a(n ; R ) are polynomials in A[R ] of bounded
with the property for
degree.
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
21 / 27
Triple products families
give a recent example of families on the algebraic group of higher rank. This aspect of the
project is studied by S. Böcherer and A. Panchishkin [Boe-Pa2006]. The triple product with
Dirichlet character χ is dened as a complex L-function (Euler
Q
L(f1 ⊗ f2 ⊗ f3 , s , χ) = p-N L((f1 ⊗ f2 ⊗ f3 )p , χ(p )p −s ), where
L((f1 ⊗ f2 ⊗ f3 )p , X )−1 = det
18
−X
(1)
αp,1
0
0
αp,1
(2)
product of degree 8)
(1)
!
⊗
0
0
αp,2
(2)
(1)
!
αp,2
⊗
αp,3
0
0
αp,3
(2)
!!
.
L-function (see [De79], [Co], [Co-PeRi]) :
Λ(f1 ⊗ f2 ⊗ f3 , s , χ) = ΓC (s )ΓC (s − k3 + 1)ΓC (s − k2 + 1)ΓC (s − k1 + 1)L(f1 ⊗ f2 ⊗ f3 , s , χ),
−s Γ(s ). The Gamma-factor determines the critical values
where ΓC (s ) = 2(2π)
s = k1 , . . . , k2 + k3 − 2 of Λ(s ), which we explicitly evaluate (like in the classical formula
2
ζ(2) = π6 ). A functional equation of Λ(s ) has the form: s 7→ k1 + k2 + k3 − 2 − s .
We use the corresponding normalized
Let us consider the product of three eigenvalues:
(1)
(1)
(1)
λ = λ(k1 , k2 , k3 ) = αp,1 (k 1 )αp,2 (k 2 )αp,3 (k3 ) with the slope
σ = vp (λ(k1 , k2 , k3 )) = σ(k1 , k2 , k3 ) = σ1 + σ2 + σ3 constant and positive
for all triplets (k1 , k2 , k3 ) in an appropriate p -adic neighbourhood of the
xed triplet of weights (k1 , k2 , k3 ).
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
22 / 27
The statement of the problem
for triple products is following: given three p-adic analytic families fj of
slope σj ≥ 0, to construct a four-variable p-adic L-function attached to
Garrett's triple product of these families. We show that this function
interpolates the special values (s , k1 , k2 , k2 ) 7−→ Λ(f1,k1 ⊗ f2,k2 ⊗ f3,k3 , s , χ)
at critical points s = k1 , . . . , k2 + k3 − 2 for balanced weights
k1 ≤ k2 + k3 − 2; we prove that these values are algebraic numbers after
dividing by certain periods. However the construction uses directly
modular forms, and not the
L-values
in question, and a comparison of
special values of two functions is done
Alexei PANCHISHKIN (Grenoble)
L-Functions of
after the construction.
Siegel Modular Forms
Oberwolfach 2007
23 / 27
Main result for triple products
Lf : (s , k1 , k2 , k3 ) 7→
1) The function
hf0 ,E(−r ,χ)i
depends
hf0 ,f0 i
p -adic
analytically
on four variables
(χ · ypr , k1 , k2 , k3 ) ∈ X × B1 × B2 × B3 ;
p -adic values:
2) Comparison of complex and
for all
(k1 , k2 , k3 )
in an anoid
neighborhood
B = B 1 × B2 × B 3 ⊂ X 3 ,
s = k2 + k3 − 2 − r
satisfying
k 1 ≤ k2 + k3 − 2:
the values at
coincide with the normalized critical special values
L∗ (f1,k1 ⊗ f2,k2 ⊗ f3,k3 , k2 + k3 − 2 − r , χ) (r = 0, . . . , k2 + k3 − k1 − 2) ,
χ mod Np v , v ≥ 1.
x ∈ X : let H = [2ordp (λ)] + 1. For any xed
(k1 , k2 , k3 ) ∈ B and x = χ · ypr then the following linear form (representing
for Dirichlet characters
3) Dependence on
modular symbols for triple modular forms)
D
x 7−→
extends to a
x ∈ X.
p -adic
f0 , E(−r , χ)
E
D
f0 , f0 ,
E
analytic function of type
Alexei PANCHISHKIN (Grenoble)
L-Functions of
o (logH (·))
of the variable
Siegel Modular Forms
Oberwolfach 2007
24 / 27
A general program
We plan to extend this construction to other sutuations as follows :
1) Construction of modular distributions
modular tower
M(ψ).
Φj
with values in an innite dimensional
2) Application of a canonical projector of type
Mα (ψ)
of
πα
onto a nite dimensional subspace
Mα (ψ).
3) General admissibility criterium. The family of distributions
Mα (ψ)
give a
h-admissible
4) Application of a linear form
µj = `(πα (Φj )),
forms πα (Φj ).
measure
`
πα (Φj )
with value in
with value in the moduli of nite rank.
of type of a modular symbol produces distributions
and an admissible measure from congruences between modular
5) One shows that certain integrals
L-values;
Φ̃
µj (χ)
of the distributions
µj
coincide with certain
however, these integrals are not necessary for the construction of measures
(already done at stage 4).
6) One shows a result of uniqueness for the constructed
h-admissible
measures : they
are determined by many of their integrals over Dirichlet characters (not all).
7) In most cases we can prove a functional equation for the constructed measure
(using the uniqueness in 6), and using a functional equation for the
L-values
µ
(over
complex numbers, computed at stage 5).
This strategy is already applicable in various cases.
Alexei PANCHISHKIN (Grenoble)
L-Functions of
Siegel Modular Forms
Oberwolfach 2007
25 / 27
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