Explicit formulas for Hecke operators and Rankin's lemma in higher genus

0.31 MB·English

by  Alexei Panchishkin

Expli
it formulas for He
ke operators and Rankin's
lemma in higher genus
7 Alexei Pan
hishkin, Kirill Vankov
0
http://www-fourier.ujf-grenoble.fr/(cid:3)pan
hish
0 @
2 e-mail : pan
hish mozart.ujf-grenoble.fr, FAX: [phone]

n
a
J
Abstra
t

5
We developexpli
it formulas for He
ke operators of higher genus in terms of

spheri
al
oordinates. Appli
ationsaregivento summationofvariousgenerating series
]
T with
oe(cid:30)
ients in lo
al He
ke algebra and in a tensor produ
t of su
h algebras. In
N parti
ular, we formulateaGnSdpp2r×ovGeSRpa2nkinG'sSlepm4main genus two. Anappli
ation to a
holomorphi
lifting from to is given usingIkeda-Miyawaki
.
h
onstru
tions.
t
a
m
Contents
[

5 1 Introdu
tion: generating series for the He
ke operators 1
v
7 2 Results 3
T(pδ)
1
2.1 Preparation: a formula for the total He
ke operator of genus 2 . . . . . 3
4
2.2 Rankin's generating series in genus 2 . . . . . . . . . . . . . . . . . . . . . . . 3
0
1 2.3 Symmetri
square generating series in genus 2 . . . . . . . . . . . . . . . . . . 5
6 2.4 Cubi
generating series in genus 2 . . . . . . . . . . . . . . . . . . . . . . . . 6
0
Sp
/ 3 Proofs: formulas for the He
ke operators of g 6
h Ω
t 3.1 Satake's spheri
al map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
a
3.2 Use of Andrianov's generating series in genus 2 . . . . . . . . . . . . . . . . . 7
m
3.3 Rankin's Lemma of genus 2 (
ompare with [Jia96℄) . . . . . . . . . . . . . . . 7
:
v L Sp
i 4 Appli
ations to -fun
tions and motives for n 8
X
GSp GSp GSp
r 5 A holomorphi
lifting from 2× 2 to 4: a
onje
ture 10
a
R(X) S(X)
A Appendix: Coe(cid:30)
ients of the polynomials and 12
1 Introdu
tion: generating series for the He
ke
operators
p p
Let be a prime. The Satakeisomorphism [Sa63℄ relates -lo
alHe
kealgebrasof redu
tive
Q
groups over to
ertain polynomial rings. Then one
an use a
omputer in order to (cid:28)nd
interesting identities between He
ke operators, between their eigenvalues, and relations to
Fourier
oe(cid:30)
ients of modular forms of higher degree.
1
The purpose of the present paper is to extend Rankin's Lemma to the summation of He
ke
series of higher genus using symboli

omputations. We refer to [Ma-Pa77℄, where Rankin's
Lemma was used in the ellipti
modular
ase for multipli
ative and additive
onvolutions of
Diri
hlet series. That work was further developped in [Pa87℄, [Pa02℄, see also [Ma-Pa05℄.
L G Q
Re
all that a
lassi
al method to produ
e -fun
tions for an algebrai
group over uses
the generating series ∞ ∞
λ (n)n−s = λ (pδ)p−δs,
X f Y X f
n=1 p primesδ=0
f G
of the eigenvalues of He
ke operators on an automorphi
form on . We study the
T(n) Sp
g
generating series of He
ke operators for the symple
ti
group , and
λ (n)=λ (T(n))
f f
.
Γ=Sp (Z) SL (Z) g
Let g ⊂ 2g be the Siegel modular group of genus , and
[p] =pI =T(p, ,p) Sp
g 2g g
··· be the s
alarHe
ke operator for . A

ording to He
ke and
2g
| {z }
Shimura,
∞
D (X)= T(pδ)Xδ
p X
δ=0
1
 , g =1
1 T(p)X +p[p] X2 if
= − 1 (see [He
ke℄, and [Shi71℄, Theorem 3.21),
1 p2[p] X2
2
−
1−T(p)X +{pT1(p2)+p(p2+1i)f[gp]=2}2X(2s−eep[3S[hpi]623T℄,(pT)hXeo3r+emp62[p),]22X4
T(p) T (p2) i=1, ,g g+1
i
where , ( ··· ) are the generatorsof the
orresponding He
ke ring
Z Sp T (p2)=[p]
g g g
over for the symple
ti
group , in parti
ular, .
In the present paper we study expli
it formulas for He
ke operatorsin higher genus in
terms of spheri
al
oordinates. Appli
ations are given to summation of various generating
series with
oe(cid:30)
ients in lo
al He
ke algebras. The question of
omputing these series
expli
itely was raised by Prof. S.Friedberg during (cid:28)rst author's talk at the
onferen
e "Zeta
Fun
tions" (The Independent Mos
ow University, September 18-22, 2006).
In parti
ular, we formulate and proveRankin's lemma in genus two for generating series
with
oe(cid:30)
ients in a tensor produ
t of lo
al He
ke algebras.
2
2 Results
T(pδ)
2.1 Preparation: a formula for the total He
ke operator of
genus 2
x ,x ,x
0 1 2
We establish (cid:28)rst the following useful formula (in spheri
al variables ):
Ω(2)(T(pδ))=p−1xδ(px(3+δ)x px(2+δ) px(3+δ)x(2+δ)+px(2+δ)x(3+δ)
x 0 1 2− 1 − 1 2 1 2 (1)
px x(3+δ)+px(2+δ)+px px x(2+δ)x2+x(1+δ)x
− 1 2 2 1− 2− 1 2 1 2
+x(2+δ)x(1+δ) x(1+δ)x(2+δ)+x2x(2+δ) x x(1+δ) x2x +x x2)/
1 2 − 1 2 1 2 − 1 2 − 1 2 1 2
((1 x )(1 x )(1 x x )(x x ))
1 2 1 2 1 2
− − − −
= p−1xδ((1 x x )(px x )x(δ+1)+(1 x x )(x px )x(δ+1)
− 0 − 1 2 1− 2 1 − 1 2 1− 2 2
(1 px x )(x x )(x x )(δ+1) (p x x )(x x ))/
1 2 1 2 1 2 1 2 1 2
− − − − − −
((1 x )(1 x )(1 x x )(x x )).
1 2 1 2 1 2
− − − −
Andrianov's generating series
The expression (1)
omes from the following Andrianov's generating series
∞ 1 x02x1x2 X2
Ω(2)(T(pδ))Xδ = − p
X x (1 x X)(1 x x X)(1 x x X)(1 x x x X)
0 0 1 0 2 0 1 2
δ=0 − − − −
after developing and a simpli(cid:28)
ation using
hange of summation.
Note that the formula (1) makes it possible to treat higher generatingseriesof the following
type ∞
D (X)= Ω(2)(T(pmδ))Xδ (m=2,3, )
p,m X x ···
δ=0
x ,x ,x
0 1 2
(in spheri
al variables ).
2.2 Rankin's generating series in genus 2
x ,x ,x y ,y ,y
0 1 2 0 1 2
Let us use the spheri
al variables and for the He
ke operators.
Note that there are two types of
onvolutions: the (cid:28)rst one is de(cid:28)ned through the Fourier
L

oe(cid:30)
ients (it was used by [An-Ka℄ for the analyti

ontinuation of the standard -
fun
tion), and the se
ond one is de(cid:28)ned through the eigenvalues of He
ke operators, and it
L
is more suitable in order to treat the -fun
tions atta
hed to tensor produ
ts of
representations of the Langlands group. However, a link between the two types is known
g =1
only for .
In orderto state a multipli
ative analogueof Rankin's lemma in genus twowe need to write
T(pδ) y ,y ,y
0 1 2
the
orrseponding formula for He
ke operator (in spheri
al variables ).
Ω(2)(T(pδ))=p−1yδ(py(3+δ)y py(2+δ) py(3+δ)y(2+δ)+py(2+δ)y(3+δ)
y 0 1 2− 1 − 1 2 1 2
py y(3+δ)+py(2+δ)+py py y(2+δ)y2+y(1+δ)y
− 1 2 2 1− 2− 1 2 1 2
+y(2+δ)y(1+δ) y(1+δ)y(2+δ)+y2y(2+δ) y y(1+δ) y2y +y y2)/
1 2 − 1 2 1 2 − 1 2 − 1 2 1 2
((1 y )(1 y )(1 y y )(y y )).
1 2 1 2 1 2
− − − −
3
Then we have that the produ
t of the above polynomials is given by
Ω(2)(T(pδ)) Ω(2)(T(pδ))=
x · y
p−2xδyδ(px(3+δ)x px(2+δ) px(3+δ)x(2+δ)+px(2+δ)x(3+δ)
0 0 1 2− 1 − 1 2 1 2
px x(3+δ)+px(2+δ)+px px x(2+δ)x2+x(1+δ)x
− 1 2 2 1− 2− 1 2 1 2
+x(2+δ)x(1+δ) x(1+δ)x(2+δ)+x2x(2+δ) x x(1+δ) x2x +x x2)
1 2 − 1 2 1 2 − 1 2 − 1 2 1 2
(py(3+δ)y py(2+δ) py(3+δ)y(2+δ)+py(2+δ)y(3+δ)
× 1 2− 1 − 1 2 1 2
py y(3+δ)+py(2+δ)+py py y(2+δ)y2+y(1+δ)y
− 1 2 2 1− 2− 1 2 1 2
+y(2+δ)y(1+δ) y(1+δ)y(2+δ)+y2y(2+δ) y y(1+δ) y2y +y y2)/
1 2 − 1 2 1 2 − 1 2 − 1 2 1 2
((1 x )(1 x )(1 x x )(x x )(1 y )(1 y )(1 y y )(y y )).
1 2 1 2 1 2 1 2 1 2 1 2
− − − − − − − −
We wish to
ompute the generating series
∞
Ω(2)(T(pδ)) Ω(2)(T(pδ))Xδ Q[x ,x ,x ,y ,y ,y ][[X]].
X x · y ∈ 0 1 2 0 1 2
δ=0
The answer is given by the following multipli
ative analogue of Rankin's lemma in genus
two:
Theorem 2.1 The following equality holds
∞
Ω(2)(T(pδ)) Ω(2)(T(pδ))Xδ =
X x · y (2)
δ=0
(px x )(1 py y )x y y
1 2 1 2 1 1 2
− −
− p2(1 x )(1 x )(x x )(1 y )(1 y )(1 y y )(1 x x y y y X)
1 2 1 2 1 2 1 2 0 1 0 1 2
− − − − − − −
x y (x px )(py y )
2 1 1 2 1 2
+ − −
p2(1 x )(1 x )(x x )(1 y )(1 y )(y y )(1 x x y y X)
1 2 1 2 1 2 1 2 0 2 0 1
− − − − − − −
x y (x px )(y py )
2 2 1 2 1 2
+ − −
p2(1 x )(1 x )(x x )(1 y )(1 y )(y1 y )(1 x y x y X)
1 2 1 1 1 2 2 0 0 2 2
− − − − − − −
x y y (x px )(1 py y )
2 1 2 1 2 1 2
− −
− p2(1 x )(1 x )(x x )(1 y )(1 y )(1 y y )(1 x x y y y X)
1 2 1 2 1 2 1 2 0 2 0 1 2
− − − − − − −
x (px x )(p y y )
1 1 2 1 2
− −
− p2(1 x )(1 x )(x x )(1 y )(1 y )(1 y y )(1 x x y X)
1 2 1 2 1 2 1 2 0 1 0
− − − − − − −
x x y (1 px x )(py y )
1 2 1 1 2 1 2
− −
− p2(1 x )(1 x )(1 x x )(1 y )(1 y )(y y )(1 x x x y y X)
1 2 1 2 1 2 1 2 0 1 2 0 1
− − − − − − −
x x y (1 px x )(y py )
1 2 2 1 2 1 2
− −
− p2(1 x )(1 x )(1 x x )(1 y )(1 y )(y y )(1 x x x y y X)
1 2 1 2 1 2 1 2 0 1 2 0 2
− − − − − − −
y y (p x x )(1 py y )
1 2 1 2 1 2
+ − −
p2(1 x )(1 x )(1 x x )(1 y )(1 y )(1 y y )(1 x y y y X)
1 2 1 2 1 2 1 2 0 0 1 2
− − − − − − −
x x (1 px x )(p y y )
1 2 1 2 1 2
+ − −
p2(1 x )(1 x )(1 x x )(1 y )(1 y )(1 y y )(1 x x x y X)
1 2 1 2 1 2 1 2 0 1 2 0
− − − − − − −
4
x y (px x )(py y )
1 1 1 2 1 2
− −
− p2(1 x )(1 x )(x x )(1 y )(1 y )(y y )(1 x x y y X)
1 2 1 2 1 2 1 2 0 1 0 1
− − − − − − −
x y (px x )(y py )
1 2 1 2 1 2
+ − −
p2(1 x )(1 x )(x x )(1 y )(1 y )(y y )(1 x x y y X)
1 2 1 2 1 2 1 2 0 1 0 2
− − − − − − −
x (x px )(p y y )
2 1 2 1 2
− −
− p2(1 x )(1 x )(x x )(1 y )(1 y )(1 y y )(1 x x y X)
1 2 1 2 1 2 1 2 0 2 0
− − − − − − −
x x y y (1 px x )(1 py y )
1 2 1 2 1 2 1 2
+ − −
p2(1 x )(1 x )(1 x x )(1 y )(1 y )(1 y y )(1 x x x y y y X)
1 2 1 2 1 2 1 2 0 1 2 0 1 2
− − − − − − −
(p x x )(p y y )
1 2 1 2
+ − −
p2(1 x )(1 x )(1 x x )(1 y )(1 y )(1 y y )(1 x y X)
1 2 1 2 1 2 1 2 0 0
− − − − − − −
y (p x x )(py y )
1 1 2 1 2
− −
− p2(1 x )(1 x )(1 x x )(1 y )(1 y )(y y )(1 x y y X)
1 2 1 2 1 2 1 2 0 0 1
− − − − − − −
y (p x x )(y py )
2 1 2 1 2
− −
− p2(1 x )(1 x )(1 x x )(1 y )(1 y )(y y )(1 x y y X)
1 2 1 2 1 2 1 2 0 0 2
− − − − − − −
Remark 2.2 (On the denominator of (2)) One (cid:28)nds using a
omputer that the
X
polynomials not depending on in the denominators of (2)
an
el after the simpli(cid:28)
ation
Q[x ,x ,x ,y ,y ,y ][[X]]
0 1 2 0 1 2
in the ring , so that the
ommon denominator be
omes
(1 x y X)(1 x y x X)(1 x y y X)(1 x y x X)(1 x y y X)
0 0 0 0 1 0 0 1 0 0 2 0 0 2
− − − − −
(1 x y x y X)(1 x y x x X)(1 x y x y X)(1 x y y x X)
0 0 1 1 0 0 1 2 0 0 1 2 0 0 1 2
− − − −
(1 x y y y X)(1 x y x y X)(1 x y x y x X)(1 x y x y y X)
0 0 1 2 0 0 2 2 0 0 1 1 2 0 0 1 1 2
− − − −
(1 x y x x y X)(1 x y y x y X)(1 x y x y x y X).
0 0 1 2 2 0 0 1 2 2 0 0 1 1 2 2
− − −
g =1
Remark 2.3 (Comparison with ) It turns out by dire
t
omputation that the
1 x2y2x y x y X2
numerator of the rational fra
tion (2) is a produ
t of the fa
tor − 0 0 1 1 2 2 by a
12 X Q[x ,x ,x ,y ,y ,y ]
0 1 2 0 1 2
polynomial of degree in with
oe(cid:30)
ients in with the
onstant
term equal to 1 and the leading term
x12y12x6x6y6y6
0 0 1 2 1 2X12.
p2
X
Moreover, the fa
tor of degree 12 does not
ontain terms of degree 1 and 11 in . The
X g =1
fa
tor of degree 2 in is very similar to one in the
ase (this series was studied and
used in [Ma-Pa77℄ ):
∞ ∞ xδ(1 x(1+δ)) yδ(1 y(1+δ))
Ω(1)(T(pδ)) Ω(1)(T(pδ))Xδ = 0 − 1 0 − 1 Xδ
X x · y X 1 x · 1 y
1 1
δ=0 δ=0 − −
1 y
1
=
(1 x )(1 y )(1 x y X) − (1 x )(1 y )(1 x y y X)
1 1 0 0 1 1 0 0 1
− − − − − −
x x y
1 1 1
+
− (1 x )(1 y )(1 x y x X) (1 x )(1 y )(1 x y x y X)
1 1 0 0 1 1 1 0 0 1 1
− − − − − −
1 x2y2x y X2
= − 0 0 1 1 .
(1 x y x y X)(1 x y x X)(1 x y y X)(1 x y X)
0 0 1 1 0 0 1 0 0 1 0 0
− − − −
2.3 Symmetri
square generating series in genus 2
Using the same method, one
an evaluate the symmetri
square generating series and the

ubi
generating series of higher genus. Note that this series, written here in spheri
al
x ,x ,x
0 1 2
variables is di(cid:27)erent from one studied by Andrianov-Kalinin, and has the form:
5
∞
Ω(2)(T(p2δ))Xδ =
X x
δ=0
x2x x
(1+x2x X+x2x X+2x2x x X+x2x x2X +x2x2x X+x4x2x2X2)(1 0 1 2X)
0 1 0 2 0 1 2 0 1 2 0 1 2 0 1 2 − p .
(1 x2x2x2X)(1 x2x2X)(1 x2x2X)(1 x2X)
− 0 1 2 − 0 1 − 0 2 − 0
2.4 Cubi
generating series in genus 2
x ,x ,x
0 1 2
The
ubi
generating series of higher genus, written here in spheri
al variables has
the form:
∞
Ω(2)(T(p3δ))Xδ =p−1( p+x6x4x2X2+x6x2x4X2+2x6x2x3X2
X x − 0 1 2 0 1 2 0 1 2
δ=0
px6x4x4X2 px6x2x4X2 2px3x2x X +x6x x3X2
− 0 1 2 − 0 1 2 − 0 1 2 0 1 2
+x6x3x X2+x6x3x5X2+x6x5x3X2+3x6x3x3X2
0 1 2 0 1 2 0 1 2 0 1 2
+x6x2x2X2+2x6x3x2X2 px3x2X px3x2X px6x4x2X2
0 1 2 0 1 2 − 0 1 − 0 2 − 0 1 2
2px3x x2X px6x2x2X2+x3x2x2X +x3x x X
− 0 1 2 − 0 1 2 0 1 2 0 1 2
px6x2x3X2 px6x3x2X2 px3x2x3X px3x3x2X
− 0 1 2 − 0 1 2 − 0 1 2 − 0 1 2
2px3x2x2X px3x3x X+x3x2x X +x9x4x4X3
− 0 1 2 − 0 1 2 0 1 2 0 1 2
2px6x3x3X2 2px3x x X +x9x4x5X3+x9x5x4X3
− 0 1 2 − 0 1 2 0 1 2 0 1 2
px6x3x4X2+x3x x2X +x9x5x5X3+x6x4x4X2
− 0 1 2 0 1 2 0 1 2 0 1 2
px6x4x3X2 px3x x3X px3x X px3x X+2x6x4x3X2
− 0 1 2 − 0 1 2 − 0 2 − 0 1 0 1 2
+2x6x3x4X2)/((1 x3X)(1 x3x3X)(1 x3x3X)(1 x3x3x3X)).
0 1 2 − 0 − 0 1 − 0 2 − 0 1 2
Sp
g
3 Proofs: formulas for the He
ke operators of
Ω
3.1 Satake's spheri
al map
Ω
Our rΩesult is basedTon(ptδh)e usLeZof the Satake spLhZe=ri
Zal[Tm(app),T,1(bpy2)a,pply,inTgn(tph2e)]spheri
al
map to elements ∈ of He
ke ring ··· for the
symple
ti
group, see [An87℄
hapter 3.
T (p2) x ,x ,x
1 0 1 2
• Case . In genus 2 (in spheri
al variables ), we obtain using
Andrianov's formulas:
x2((x2x +x x2)p2+x x p2 x x +(x +x )p2)
Ω(T (p2))= 0 1 2 1 2 1 2 − 1 2 1 2 .
1 p3
T (p2)=[p] T(p)
2 2
• Cases and
x2x x
Ω(T (p2))= 0 1 2, Ω(T(p))=x (1+x )(1+x ).
2 p3 0 1 2
6
3.2 Use of Andrianov's generating series in genus 2
We refer to [An87℄, p.164, (3.3.75) for the following
elebrated summation formula:
∞ 1 x02x1x2 X2
Ω(2)(T(pδ))Xδ = − p ,
X (1 x X)(1 x x X)(1 x x X)(1 x x x X) (3)
0 0 1 0 2 0 1 2
δ=0 − − − −
whi
h gives after development and simpli(cid:28)
ation the following formula
Ω(2)(T(pδ))=p−1xδ(px(3+δ)x px(2+δ) px(3+δ)x(2+δ)+px(2+δ)x(3+δ)
0 1 2− 1 − 1 2 1 2
px x(3+δ)+px(2+δ)+px px x(2+δ)x2+x(1+δ)x
− 1 2 2 1− 2− 1 2 1 2
+x(2+δ)x(1+δ) x(1+δ)x(2+δ)+x2x(2+δ) x x(1+δ) x2x +x x2)/
1 2 − 1 2 1 2 − 1 2 − 1 2 1 2
((1 x )(1 x )(1 x x )(x x )).
1 2 1 2 1 2
− − − −
x ,...,x y ,...,y Ω Ω
0 n 0 n x y
Then we use two groups of variables: and in two
opies , of the
speri
al map, in order to treat the tensor produ
t of two lo
al He
ke algebras.
Next, in order to
arry out the summation of the series
∞
Ω(2)(T(pδ)) Ω(2)(T(pδ))Xδ
X x · y
δ=0
δ
on a
omputer, we used a subdivision of ea
h summand (over ) into smaller parts. These
xδ,yδ,xδ,yδ,(x x )δ,(y y )δ
parts
orrespond to symboli
monomials in 1 1 2 2 1 2 1 2 .
3.3 Rankin's Lemma of genus 2 (
ompare with [Jia96℄)
Let us
ompute the series
∞
Dp(1,1)(X)=XT(pδ)⊗T(pδ)Xδ ∈L2,Z⊗L2,Z[[X]]
δ=0
L L
2,Z 2,Z
in terms of the generatorsof He
ke's algebra ⊗ given by the following operators:
T(p) 1,T1(p2) 1,[p] 1,1 T(p),1 T1(p2),1 [p] L2,Z L2,Z[[X]].
⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ∈ ⊗
g =2
Theorem 3.1 (O
tober 2006) For , we have the following expli
it representation
∞ (1 p6[p] [p]X2) R(X)
D(1,1)(X)= T(pδ) T(pδ)Xδ = − ⊗ · ,
p X ⊗ S(X) where
δ=0
R(X),S(X) L2,Z L2,Z[X]
∈ ⊗
are given by the equalities (4) and (5):
R(X)=1+r2X2+ +r10X10+r12X12 L2,Z L2,Z[X] r1 =r11 =0,
··· ∈ ⊗ with (4)
S(X)=1+s X + +s X16
1 16
··· (5)
=1 (T(p) T(p))X + +(p6[p] [p])8X16 L2,Z L2,Z[X],
− ⊗ ··· ⊗ ∈ ⊗
r s
i i
with and given in Appendix. Moreover, there is an easy fun
tional equation (similar to
[An87℄, p.164, (3.3.79)):
s =(p6[p] [p])8−is (i=0, ,8).
16−i i
⊗ ···
7
g =1
Remark 3.2 (Comparison with the
ase (in terms of He
ke operators))
g =1
The
orresponding result in the
ase written in terms of He
ke operators, looks as
follows (see 2.3):
∞
T(pδ) T(pδ)Xδ =(1 p2[p] [p]X2)/
X ⊗ − ⊗
δ=0
(1 T(p) T(p)X + p(T(p)2 [p]+[p] T(p)2) 2p2[p] [p] X2
− ⊗ (cid:0) ⊗ ⊗ − ⊗ (cid:1)
p2T(p)[p] T(p)[p]X3+p4[p]2 [p]2X4).
− ⊗ ⊗
Indeed this follows dire
tly from Remark 2.3
L Sp
n
4 Appli
ations to -fun
tions and motives for
The Fourier expansion of a Siegel modular form.
f = a(T)qT Mn k n
Let X ∈ k be a Siegel modular form of weight and of genus on the
T∈Bn
H = z M (C) Im(z)>0
n n
Siegel upper-half plane { ∈ | }.
f
The formal Fourier expansion of uses the symbol
n
qT =exp(2πitr(Tz))= qTii q2Tij
Y ii Y ij
i=1 i<j
C[[q ,...,q ]][q , q−1] q =exp(2π(√ 1z )) T
∈ 11 nn ij ij i,j=1,···,m, where ij − i,j , and is in the
B = T =tT 0T .
n
semi-group { ≥ | half-integral}
Satake parameters of an eigenfun
tion of He
ke operators
f Mn f f T T L
Suppose that ∈ k is an eigenfun
tion of all He
ke operators 7−→ | , ∈ n,p for all
p f T =λ (T)f
f
primes , hen
e | λ (T) .C λ :L C
f f n,p
T(nh+en1a)ll the numbers ∈ d(eα(cid:28)n,eαa,hom,oαm)or=ph(iCsm×)n+1 −→ given by a f
0 1 n
-tuαp2lαe of
oαmp=lepxknnu−mn(bne+r1s)/2. ··· (the Satake parameters of ).
One has 0 1··· n g Mn
For another Siegel modular formλ,:eiLgenfun
tiCon of He
ke operators ∈ l
onsider the
g n,p
(
βorr,eβsp,ondi,nβg)homgomorphismλ λ :L−→ Lgiven byCits Satake parameters
0 1 n f g n,p n,p
··· of , and let ⊗ ⊗ −→ .
L Sp
-fun
tions, fun
tional equation and motives for n (see [Pa94℄, [Yosh01℄)
One de(cid:28)nes
n
Q (X)=(1 α X) (1 α α α X),
• f,p − 0 Y Y − 0 i1··· ir
r=11≤i1<···<ir≤n
n
R (X)=(1 X) (1 α−1X)(1 α X) Q[α±1, ,α±1][X].
• f,p − Y − i − i ∈ 0 ··· n
i=1
L L(Sp(f),s,χ) L L(St(f),s,χ) f
Then the spinor -fun
tion and the standard -fun
tion of
s C χ
(for ∈ , and for all Diri
hlet
hara
ters ) are de(cid:28)ned as the Euler produ
ts
8
L(Sp(f),s,χ)= Q (χ(p)p−s)−1
• Y f,p
p
L(St(f),s,χ)= R (χ(p)p−s)−1
• Y f,p
p
L
Motivi
-fun
tions
k >n
Following [Pa94℄ and [Yosh01℄, these fun
tions are
onje
tured to be motivi
for all :
L(Sp(f),s,χ)=L(M(Sp(f))(χ),s), L(St(f),s,χ)=L(M(St(f))(χ),s),
M(Sp(f)) M(St(f)) f
and the motives and are pure if is a genuine
usp form (not
oming
from a lifting of a smaller genus):
M(Sp(f)) Q Q(λf(n))n∈N 2n
• is a motive over with
oe(cid:30)
ients in of rank , of weight
w=kn n(n+1)/2 Hp,q
p,q
− , and of Hodge type ⊕ , with
p=(k i )+(k i )+ +(k i ),
1 2 r
− − ··· − (6)
q =(k j )+(k j )+ +(k i ), r+s=n,
1 2 s
− − ··· − where
1 i <i < <i n,1 j <j < <j n,
1 2 r 1 2 s
≤ ··· ≤ ≤ ··· ≤
i , ,i j , ,i = 1,2, ,n ;
1 r 1 s
{ ··· }∪{ ··· } { ··· }
M(St(f)) Q Q(λf(n))n∈N 2n+1
• w=is0a motive over withH
o0,e0(cid:30)
nient(sHin−k+i,k−i Hk−oi,f−rka+ni)k , of
weight , and of Hodge type ⊕i=1 ⊕ .
A fun
tional equation
L L
Following general Deligne's
onje
ture [De79℄ on the motivi
-fun
tions, the -fun
tion
satisfy a fun
tional equation determined by the Hodge stru
ture of a motive:
Λ(Sp(f),kn n(n+1)/2+1 s)=ε(f)Λ(Sp(f),s),
− − where
Λ(Sp(f),s)=Γ (s)L(Sp(f),s),ε(f)=( 1)k2n−2,
n,k
−
Γ1,k(s)=ΓC(s)=2(2π)−sΓ(s) Γ2,k(s)=ΓC(s)ΓC(s k+2)
Γn,k(s)=Qp<qΓC(s−p)ΓaR+(s,−(w/2))ΓR(s+1−(w−/2))a−,foarndsome non-negativeintegers
a+ a− a++a− =w/2 ΓR(s)=π−s/2Γ(s/2)
and , with , and .
g =2
Motive of the Rankin produ
t of genus
f g k l k >l M(Sp(f))
Let and be two Siegel
usp eigenforms of weights and , , and let and
M(Sp(g)) f g M(Sp(f)) Q
be the spinor motives of and . Then is a motive over with
Q(λf(n))n∈N 4 w =2k 3
H
o0e,(cid:30)2k
−ie3ntsHink−2,k−1 Hk−1o,kf−r2ankH,2ko−f3w,0eight M(Sp(−g)), and of Hodge tyQpe
⊕ ⊕ ⊕ , and is a motive over with
Q(λg(n))n∈N 4 w =2l 3
H
o0e,(cid:30)2l−
i3entsHinl−2,l−1 Hl−1,lo−f2ranHk2l−, 3o,f0weight − , and of Hodge type
⊕ ⊕ ⊕ .
M(Sp(f)) M(Sp(g)) Q
The tensor produ
t ⊗ is a motive over with
oe(cid:30)
ients in
Q(λf(n),λg(n))n∈N 16 w =2k+2l 6
of rank , of weight − , and of Hodge type
9
H0,2k+2l−6 Hl−2,2k+l−4 Hl−1,2k+l−5 H2l−3,2k−3
⊕ ⊕ ⊕
Hk−2,k+2l−4 Hk+l−4,k+l−2 Hk+l−3,k+l−3 Hk+2l−5,k−1
⊕ ⊕ + ⊕
Hk−1,k+2l−5 Hk+l−3,k+l−3 Hk+l−2,k+l−4 Hk+2l−4,k−2
⊕ − ⊕ ⊕
H2k−3,2l−3 H2k+l−5,l−1 H2k+l−4,l−2 H2k+2l−6,0.
⊕ ⊕ ⊕
L
Motivi
-fun
tions: analyti
properties
L
Following Deligne's
onje
ture [De79℄ on motivi
-fun
tions, applied for a Siegel
usp
F Sp (Z) n=4 k >5
eigenform for the Siegel modular group 4 of genus and of weight , one
Λ(Sp(F),s)=Λ(Sp(F),4k 9 s)
has − − , where
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),
× − − −
(
ompare this fun
tional equation with that given in [An74℄, p.115).
n=2 f g Sp (Z) k,l
On the other hand, for and fortwo
usp eigenforms and for 2 of weights ,
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
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)
We used here the Gauss dupli
ation formula . Noti
e that
a =a =1 M(Sp(f)) M(Sp(g))
+ −
in this
ase, and the
onje
tural motive ⊗ does not admit

riti
al values.
GSp GSp GSp
2 2 4
5 A holomorphi
lifting from × to : a

onje
ture
(
ompare with
onstru
tions in [BFG06℄, [BFG92℄, [Jia96℄ for generi
automorphi
forms).
Our
omputation makes it possible to
ompare the spinor He
ke series of genus 4
omputed
u ,u ,u ,u ,u
0 1 2 3 4
in [VaSp4℄ (in variables ) with the Rankin produ
t of two He
ke series of
x ,x ,x ,y ,y ,y
0 1 2 0 1 2
genus 2 (in variables ). It follows from our
omputation that if we make
u =x y ,u =x ,u =x ,u =y ,u =y
0 0 0 1 1 2 2 3 1 4 2
the substitution then the denominator of the
series ∞
Ω(4)(T(pδ))Xδ
X u
δ=0

oin
ides with the denominator of the Rankin produ
t
∞
Ω(2)(T(pδ)) Ω(2)(T(pδ))Xδ Q[x ,x ,x ,y ,y ,y ][[X]].
X x · y ∈ 0 1 2 0 1 2
δ=0
On the basis of this equality we would like to push forward the following
10