Théorie des groupes/Group Theory Théorie des

Th´eorie des groupes/Group Theory
Th´eorie des Nombres/Number Theory
A FORMAL THEORY OF EISENSTEIN SERIES
Kˆ azım ˙ Ilhan ˙ Ikeda
May 1995
Abstract. The definition of the Eisenstein series E
E
E(Z, s;k, Γ) on the symplectic and uni-
tary groups Gis well-known. We introduce a series E(Z, λ) whose construction involves a
5-tuple {G, P, M, W, ²}satisfying certain conditions. We prove that, if λ:Mn×2n(k)−→ C
is a locally-constant function, then the series E(Z, λ) is a finite linear combination of G-
transforms of the Eisenstein series E
E
E(Z, s;k, Γ) whose coefficients are products of certain
Hecke L-functions.
UNE TH´ EORIE FORMELLE DES S´ ERIES d’EISENSTEIN
R´ esum´ e. Sur les groupes symplectique et unitaire Gla d´efinition des s´eries d’Eisenstein
E
E
E(Z, s;k, Γ) est bien connue. Dans cette note nous introduisons une s´erie E(Z, λ) dont la
construction fait intervenir un 5-tuplet {G, P, M, W, ²}satisfaisant `a certaines conditions.
Nous d´emontrons que, si λ:Mn×2n(k)−→ Cest une fonction localement constante, alors
la s´erie E(Z, λ) est une combinaison lin´eaire finie des G-transform´es des s´eries d’Eisenstein
E
E
E(Z, s;k, Γ) dont les coefficients sont produits de certaines fonctions Lde Hecke.
Version fran¸caise abr´eg´ee. §1. Consid´erons la donn´ee (que nous appelons une donn´ee
d’Eisenstein dans la suite) d´efinie par un 5-tuplet {G, P, M, W, }o`u ∈West un
ensemble non-vide, Get Msont des groupes de transformations sur W, et Pest un
sous-groupe de Gsatisfaisant aux conditions suivantes:
(i) Gop`ere transitivement `a droite sur Wet Mop´ere fid`element `a gauche de W. Ces
actions sur Wsont compatibles comme dans (1); (ii) T²⊂Po`u T²d´esigne le groupe
d’isotropie de dans G;(iii) Il existe un morphisme surjectif d:P−→ Mtel que le carr´e
(2) est commutatif, o`u r²est d´efini par l’action `a droite de Psur et l²est l’application
d´efinie par l’action `a gauche de Msur ;(iv) Il existe un G-espace Het un groupe
G=GL∗(C)avec un facteur d’automorphie j:G×H−→ G `a valeurs dans Gtel que la
condition (3) est satisfaite pour tous π∈Pet z∈H; o`u δ:M−→ G est un morphisme.
§2. (Tous les calculs du §2 sont formels.) Soit Γ un sous-groupe de Get soit ∆ un sous-
groupe de Mv´erifiant la condition (4). Etant donn´e un caract`ere ψ: Γ →T, v´erifiant
(7), o`u Bd´esigne un syst`eme complet de repr´esentants de P\G/Γ, on consid`ere les s´eries
E(z, λ, ψ) comme dans (8), o`u λ:W→Cest une application quelconque v´erifiant (6).
La s´erie d’Eisenstein E
E
E(z, Γ, ψ) de G(relative `a Γ et `a ψ) est d´efinie comme dans (9).
On d´emontre au §2 (Th´eor`eme 2.3) que la s´erie E(z, λ, ψ) est une combinaison lin´eaire
des β-transform´es des s´eries d’Eisenstein E
E
E(z, βΓβ−1, ψβ) pour β∈B.
This report announces some of the results of the author’s Ph.D. thesis, Princeton University 1993.
The author heartily thanks Goro Shimura for his help and encouragement all through author’s graduate
study. He also thanks the referee for his valuable suggestions.
Typeset by A
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§3. Dans cette section, Fd´esigne un corps de nombres totalement r´eel de degr´e m
et Kd´esigne un CM -corps ayant Fpour sous-corps totalement r´eel maximal. Pour
uniformiser notre expos´e, le symbole kd´esigne soit F, soit Ks’il n’y a aucune confu-
sion et Okd´esigne l’anneau des entiers de k. On note a
a
al’ensemble a
a
aFdes premiers
archim´ediens de Fou bien d’un CM-type fix´e de K, et h
h
hd´esigne l’ensemble des premiers
non-archim´ediens de F. Pour X∈Mn×s(K), nous posons X∗=tXι, o`u ι:K−→ K
est l’involution de Galois sur F, et Jn=µ0−1n
1n0¶∈M2n(Q). Nous consid´erons les
groupes alg´ebriques Sp(n, F ) et SU(n, K) d´efinis sur Fque nous noterons simplement
par Gk. Nous introduisons ∼ Gkcomme dans (12). Notons Pkle sous-groupe parabolique
de Gkcorrespondant `a la partition (n, n), Qkla composante de Levi et Rkle radi-
cal unipotent de Pk. Les sous-groupes ∼ Pk,∼ Qket ∼ Rkde ∼ Gksont d´efinis de la mˆeme
mani`ere. Soit Wk={α∈Mn×2n(k)|αJnα∗= 0,rang(α) = n}. On d´emontre que
{∼ Gk,∼ Pk, GLn(k), Wk,(0n,1n)}est une donn´ee d’Eisenstein. Pour un sous-groupe congru-
ence Γ de ∼ Gk(cf [6] pp 424), d´efinissons la s´erie d’Eisenstein E
E
E(Z, s;k, Γ) comme dans
(15), en supposant que Qν∈a
a
adet(µν(xν, zν))k= 1 pour tout x∈Pk∩Γ pour que la somme
ait un sens. Soit ∆ un sous-groupe de congruence de GLn(k) suffisamment petit pour
que δ(∆) = 1. Notons par S(V) l’espace des fonctions localement constantes d´efinies sur
un espace vectoriel Vsur Q`a valeurs dans C. Consid´erons un ´el´ement λ∈ S(Mn×2n(k)),
tel que λ(∆wΓ) = λ(w) pour w∈Mn×2n(k). Suivant la notation de (5), introduisons la
somme E(Z, λ) comme dans (16).
Th´eor`eme 3.1. Soit Γun sous-groupe de congruence de ∼ Gksatisfaisant `a la condition
Qν∈a
a
adet(µν(xν, zν))k= 1 pour tout x∈Pk∩Γet tout Z= (zν)ν∈a
a
a∈Ha
a
aet soit ∆
un sous-groupe de congruence de GLn(k)tel que δ(∆) = 1. Soit λ:Mn×2n(k)−→ C
une fonction localement constante avec la propri´et´e λ(∆wΓ) = λ(w)pour w∈Mn×2n(k).
Alors, la s´erie E(Z, λ)est une combinaison lin´eaire finie des β-transform´es E
E
E(Z, s :
k, βΓβ−1)de la s´erie d’Eisenstein dont les coefficients sont des sommes de produits de
s´eries Lde Hecke.
1.PRELIMINARIES. Consider the data (which we call an Eisenstein datum in the
sequel) defined by a 5-tuple {G, P, M, W, }, where ∈Wis a (non-empty) set, Gand M
transformation groups on Wand Pa subgroup of Gsatisfying the following conditions:
(i) Gacts transitively on the right of Wand Macts faithfully on the left of W. These
actions on Ware compatible in the following sense:
(1) (mw)g=m(wg),for every m∈M,w∈Wand g∈G.
(ii) T²⊂P, where T²denote the isotropy group of in G.
(iii) There exist a surjective morphism d:P−→ Msuch that the square
(2)
Pr²
−−−−→ W
idP
yx l²
Pd
−−−−→ M
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is commutative, where r²:P−→ Wis defined via the right action of Pon and
l²:M−→ Wis the map defined by the left action of Mon .
(iv) There exists a G-space Hand a group G=GL∗(C)with a G-valued factor of auto-
morphy j:G×H−→ G (that is the cocycle relation j(g1g2, z) = j(g1, g2(z))j(g2, z), for
g1, g2∈Gand for z∈His satisfied) such that
(3) j(π, z) = δ(d(π)),
for every π∈Pand z∈H; where δ:M−→ G is a morphism.
Let {G, P, M, W, }be an Eisenstein datum. Define the mapping τ:G→Wby
g7→ τ(g) = g, for g∈G. Then we readily have the following
Lemma 1.1. (i) The map τ:G−→ Wis a surjection with τ(1) = ,τ(g1g2) = τ(g1)g2
and τ(πg) = d(π)τ(g)for every g, g1, g2∈Gand π∈P;(ii) T²=Ker(d);(iii) j(g, z) =
1for every g∈T²and z∈H;(iv) The mapping τ:G−→ Winduces a bijection
ιτ:T²\G−→ Wdefined by T²g7−→ τ(g)for g∈G.
2.A FORMAL THEORY OF EISENSTEIN SERIES. In this paragraph all of the
computations are formal and we ignore the convergence questions. Let Γ be a subgroup of
Gand Bbe a complete set of representatives of the double-coset decomposition P\G/Γ of
Gwith respect to (P, Γ). That is, there is the disjoint union G=Fβ∈BP βΓ. For β∈B,
P∩βΓβ−1becomes a transformation group on the set βΓ, where the action of P∩βΓβ−1
on βΓ is defined via left multiplication. Let Sβbe a complete set of representatives of
the orbits in βΓ relative to P∩βΓβ−1. The proof of the following Lemma is elementary,
so it is omitted.
Lemma 2.1. (i) G=Fβ∈BP Sβ;(ii) The set Wis the disjoint union of the sets M Sβ
and the map defined by (m, x)7→ mx is a bijection from Fβ∈BM×Sβto W.
Let ∆ be a subgroup of M(then ∆ is a transformation group on W, where the action
of ∆ on Wis induced from that of Mon W). The orbits ∆\Win Wrelative to ∆ have
the following description:
Lemma 2.2. ∆\W=Fβ∈B{rx |r∈∆\M, x ∈Sβ}.
Assume that
(4) d(P∩βΓβ−1)⊂∆⊂Ker(δ),
for every β∈B. So there is the canonical induced mapping δ∗: ∆\M−→ G. As a
notation (which are clearly well-defined) introduce:
(5) j(g, z) = j(τ(g), z) and j(x, z) = j(w, z),
for g∈G,x= ∆w∈∆\Wand z∈H. Take λ:W−→ Cany map satisfying the
condition
(6) λ(∆wγ) = ψ(γ)λ(w),
for any γ∈Γ and w∈Wwith a character ψ: Γ −→ T={z∈C:|z|= 1}such that
(7) P∩βΓβ−1⊂Ker(ψ),
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for every β∈B. We now introduce the following series:
(8) E(., λ, ψ) : H−→ G defined by E(z, λ, ψ) = X
x∈∆\W
λ(x)j(x, z),
for z∈H, which is a well-defined function by virtue of the assumption (6).
(9) E
E
E(., Γ, ψ) : H−→ G defined by E
E
E(z, Γ, ψ) = X
x∈(P∩Γ)\Γ
ψ(x)j(x, z),
for z∈H, which is a well-defined function, since d(P∩Γ) ⊂∆⊂Ker(δ) and P∩Γ⊂
Ker(ψ). The series introduced in (9) is the Eisenstein series of G(with respect to Γ and
ψ: Γ −→ T). For β∈B, denote by ψβ:βΓβ−1−→ Tto be the character which factors
through
(10) ψβ:βΓβ−1β−1−conjugation
−−−−−−−−−−−→ Γψ −→ T.
Consider the Eisenstein series E
E
E(z, βΓβ−1, ψβ), which is well-defined since d(P∩βΓβ−1)⊂
∆⊂Ker(δ) and P∩βΓβ−1⊂Ker(ψ). Then for β∈B, the β-transform of E
E
E(., βΓβ−1, ψβ) :
H−→ G is given by
(11) E
E
E||β(z, βΓβ−1, ψβ) = X
x∈Sβ
ψ(β−1x)j(x, z),for z∈H.
We now state the main theorem: the series E(z, λ, ψ) is a linear combination of β-
transforms of Eisenstein series E
E
E(z, βΓβ−1, ψ) for β∈B. More precisely,
Theorem 2.3. For any z∈H,E(z, λ, ψ) = Pβ∈BLβE
E
E||β(z, βΓβ−1, ψβ)where Lβ=
Pr∈∆\Mλ(rβ)δ∗(r)for β∈B.
3. APPLICATION OF THE FORMAL THEORY OF EISENSTEIN SE-
RIES. Let Fbe a totally real algebraic number field of degree mand KaCM-field
with the maximal totally real subfield F(i.e Kis a totally imaginary quadratic extension
of F). To make our exposition uniform, the symbol kwill denote either For Kif there
is no fear of confusion and Okwill denote the ring of integers of k. We denote by a
a
aeither
a
a
aFthe set of archimedean primes of For a fixed CM -type of Kand h
h
hwill denote the
set of non-archimedean primes of F. We call ιthe Galois involution of K/F . Let ∼ Gkbe
the algebraic group defined over Fby
(12) ∼ Gk={X∈GL2n(k)|XJnX∗=Jn},
where Jn=µ0−1n
1n0¶∈M2n(Q) and X∗denotes respectively, the transpose tXin
the totally real case k=F, and the transpose-conjugate tXιin the CM-case k=K.
We denote the group ∼ Gk∩SL2n(k) simply by Gk. Let Pbe the parabolic subgroup of
Gcorresponding to the partition (n, n), Qthe Levi component and Rthe unipotent
radical of P. The subgroups ∼ P,∼ Qand ∼ Rof ∼ Gare defined likewise. Let Wk={α∈
Mn×2n(k)|αJnα∗= 0,rank(α) = n}. The surjection τ:∼ Gk−→ Wkdefined by
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τ(X) = (0n,1n)X= (cX, dX) for X=µaXbX
cXdX¶∈∼
Gkinduces a right transitive action
of ∼ Gkon Wkas τ(X)Y=τ(XY ) for Xand Yelements of ∼ Gk. Let d:∼
Pk−→ GLn(k) be
the surjective homomorphism defined as d(X) = dXfor X=µaXbX
0dX¶∈∼
Pk. Then
we have the commutative square (2). The group GLn(k) acts faithfully on Wkfrom the
left by m(c, d) = (mc, md) for m∈GLn(k) and (c, d)∈Wk. The right action of ∼ Gkand
the left action of GLn(k) on Wkare clearly compatible. The isotropy subgroup T(0n,1n)
of (0n,1n) in ∼ Gkis the unipotent radical ∼ Rkof ∼ Pk. The local archimedean group ∼ Gk,ν
(ν∈a
a
a) acts on the symmetric space
Hk=½{Z∈Mn(C)|tZ=Z, Im(Z)>0},k=F,
{Z∈Mn(C)|i(tZ−Z)>0},k=K,
as α(Z) = (aαZ+bα)(cαZ+dα)−1for α∈∼ Gk,ν and Z∈Hk. The map µν:∼ Gk,ν ×Hk−→
GLn(C) defined by µν(α, z) = cαz+dαis a GLn(C)-valued factor of automorphy of ∼ Gk,ν
on Hk. Following the notation of [6], we introduce
(13) Jk,s(x, Z) = Y
ν∈a
a
a
det(µν(xν, zν))k|det(µν(xν, zν)) |s,
where Z= (zν)ν∈a
a
a∈Ha
a
a
k,x∈G →GA,k∈Zand s∈C. It is clear that
(14) Jk,s(π, Z) = Nk/Q(det(d(π)))k/[k:F]|Nk/Q(det(d(π))) |s/[k:F],for π∈∼
Pk.
So the morphism δ:GLn(k)−→ C×,δ(g) = Nk/Q(det(g))k/[k:F]|Nk/Q(det(g)) |s/[k:F]for
g∈GLn(k) satisfies δ(d(π)) = Jk,s(π, Z) for every π∈∼ Pkand Z∈Ha
a
a
k. Hence we have
shown that: {∼ Gk,∼ Pk, GLn(k), Wk,(0n,1n)}is an Eisenstein datum.
For a congruence subgroup Γ of ∼ Gk(cf [6] pp 424), define the Eisenstein series
(15) E
E
E(Z, s;k, Γ) = X
α∈(Pk∩Γ)\Γ
Jk,s(α, Z)−1
for Z= (zν)ν∈a
a
a∈Ha
a
a
k,s∈C,k∈Z; provided that Qν∈a
a
adet(µν(xν, zν))k= 1 for
every x∈Pk∩Γ to make the sum meaningful. The series E
E
E(Z, s;k, Γ) converges for
Re(k+s)>½n+ 1 (k=F)
2n(k=K). Let ∆ be a sufficiently small congruence subgroup of
GLn(k) such that δ(∆) = 1. Let S(V) denote the space of locally constant functions
V−→ Con a vector space Vover Q. Recall that a function :V→Cis called
locally constant, if there exist two lattices L1and L2of Vso that (x) = 0 if x /∈L1
and (x) = (y) if x≡y(mod L2). Consider an element λ∈ S(Mn×2n(k)), such that
λ:Mn×2n(k)−→ Csatisfies the condition λ(∆wΓ) = λ(w) for w∈Mn×2n(k). Following
the notation (5), introduce the sum
(16) E(Z, λ) = X
x∈∆\Wk
λ(x)Jk,s(x, Z)−1,
for Z∈Ha
a
a
k. Theorem 2.3 yields E(Z, λ) = Pβ∈BLβE
E
E||β(Z, s;k, βΓβ−1), where B
is a complete set of representatives of ∼ Pk\∼ Gk/Γ which is known to be a finite set and
Lβ=Pr∈∆\GLn(k)λ(rτ(β))Nk/Q(det(r))−k/[k:F]|Nk/Q(det(r)) |−s/[k:F]. Such Dirichlet
series are studied by Shimura in [7] (pp. 309-313). We state the main theorem of this
paragraph. The proof utilizes Proposition 9.2 of [7].
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