Eisenstein series on Siegel modular groups Winfried Kohnen
In this short note we would like to show how one can obtain a non-holomorphic Eisenstein series on the Jacobi group of genus g from a non-holomorphic Eisensenstein series on the Siegel modular group of genus g+ 1 by carrying out a certain limit process.
1. Notations
For a natural number g we let
Hg(R) ={((λ, µ), κ)|(λ, µ)∈Rg×Rg, κ∈R} be the Heisenberg group, with group law given by
((λ, µ), κ)((λ′, µ′), κ′) = ((λ+λ′, µ+µ′), κ+κ′+λµ′t−µλ′t).
The symplectic group Spg(R) operates on Hg(R) by ((λ, µ), κ)◦M = ((λ, µ)M, κ), and the semi-direct product
GJg =Spg(R)▷Hg(R)
operates on Hg ×Cg (with Hg the Siegel upper half-space of degree g) by γ ◦(τ, z) = ((aτ +b)(cτ +d)−1,(z +λτ +µ)(cτ +d)−1).
Given an even integerkand a positive integermit operates on functionsf :Hg×Cg → C by
(f|k,mγ)(τ, z) = det (cτ +d)−ke−2πim((cτ+d)−1c)[(z+λτ+µ)t]
·e2πim(κ+µλt+τ[λt]+2λzt)f(γ◦(τ, z)) (τ ∈ Hg, z ∈Cg, γ = (
(a b c d
)
,((λ, µ), κ))).
We let ΓJg =GJg(Z) be the Jacobi group.
2. Results
Let
Ek(g+1)(Z;s) = (detY)s ∑
GLg+1(Z)\{(C,D)=1}
det (CZ +D)−k|det (CZ+D)|−2s
1
(Z =X+iY ∈ Hg+1; s∈C, Re (s)> g−k 2 + 1)
be the non-holomorphic Siegel-Eisenstein series of weight k and genus g+ 1 (studied by Maass, Langlands, Kalinin, Shimura, Mizumoto and others).
Theorem 1. Let
Ek(g+1)(Z;s) = ∑
m∈Z
ek,m(τ, z;v′;s)e2πimu′
(Z =
(τ′ z zt τ
)
, τ′ ∈ H, z ∈Cg, τ ∈ Hg, τ′ =u′+iv′) be the Fourier-Jacobi expansion of Ek(g+1)(Z;s). Then for m squarefree one has
ek,m(τ, z;v′;s) = (−1)k/2(2π)2s+k
Γ(s+k)ζ(2s+k) ·(detY)s ∑ (a b
c d )
∈Γg,∞\Γg,λ∈Zg
det (cτ +d)−k
·|det (cτ +d)|−2s·W˜k/2,s+(k−1)/2(4πm(v′+ Imh(
(a b c d
)
, λ;τ, z)))
·exp (2πimReh(
(a b c d
)
, λ;τ, z)).
Here
Γg =Spg(Z),Γg,∞ ={
(a b c d
)
∈Γg|c= 0}, W˜α,β(w) :=w−β−1/2Wα,β(w) (w∈C, w ̸= 0), Wα,β(w) (w ̸= 0) is the Whittaker function of index (α, β) and
h(
(a b c d
)
, λ;τ, z) :=−((cτ+d)−1c)[zt] + 2λ(cτ +d)−1tzt + ((aτ +b)(cτ +d)−1)[λt].
Proof. For a detailled proof see [2]. One proceeds in a similar way as in the holomorphic case (see [1]), i.e. splits up the summation in the Eisenstein series according to rkC = ν, 0 ≤ ν ≤ g+ 1. To get suitable sets of representatives for the latter matrices one uses the usual embeddings Γ1 ,→Γg+1, GLg+1(Z),→Γg+1 etc.
The Whittaker function shows up from the Fourier expansion of the one-variable non- holomorphic Eisenstein series.
The function ek,m(τ, z;v′;s)e2πimu′ behaves like a Jacobi modular form of weight k and index m w.r.t. ΓJg ,→ Spg+1(R). However, it depends on v′, too and so cannot be considered as a “true” Jacobi form. To get around this, we put
v′ :=v−1[yt] + δ
4πm (δ >0 fixed ).
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Theorem 2. The limit
limδ→∞eδ/2e2πmv−1[yt]ek,m(τ, z;v−1[yt] + δ
4πm) (Re (s)> g−k 2 + 1) exists and (up to a simple Gamma-factor) is equal to
Ek,mJ (τ, z;s) := (detv)s ∑ (a b
c d )
∈Γg,∞\Γg,λ∈Zg
det (cτ+d)−k|det (cτ +d)|−2s
·exp (2πimReh(
(a b c d
)
, λ;τ, z))
= ∑
γ∈ΓJg,∞\ΓJg
(det Imτ)s|k,mγ (v= detτ)
(with ΓJg,∞ := {(M,((0, µ), κ))|M ∈ Γg,∞, µ ∈ Zg, κ ∈ Z}) which is a non-holomorphic Jacobi-Siegel-Eisenstein series of weight k and index m w.r.t. ΓJg (real-analytic in τ, holomorphic in z).
Remark. The above series for g= 1 were studied by Arakawa and Sugano.
Proof. This easily follows from the asymptotics ew/2w−αWα,β(w) = 1 +O( 1
|w|) (|w| → ∞).
For some details cf. [2].
Remark. In particular, if ℓ ∈N, ℓ >3 and one putsg= 1, s =ℓ− 12, k = 1−ℓ in Theorem 2, one gets the skew-holomorphic Eisenstein series (in the sense of Skoruppa) of weight ℓ and index m on ΓJ1.
One can naturally ask if a similar limiting process as above would also work for cusp forms.
References.
[1] S. B¨ocherer, ¨Uber die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen. Math.
Z. 183, 21-46 (1983)
[2] W. Kohnen, Non-holomorphic Poincar´e-type series on Jacobi groups, J. of Number theory 46, no. 170-199 (1994)
Author’s address: Mathenatisches Institut der Universit¨at, INF 288, 69120 Heidelberg, Germany
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