Let $\Gamma_n=Sp_{2n}(\mathbb{Z})$ and write $\gamma=\left(\matrix{A & B \\ C & D}\right)\in\Gamma_n$ for its block decomposition. Further let $\Gamma_n^0$ be the subset consisting of symplectic matrices such that $C=0$. Let $\mathcal{H}_n=\{Z\in\mathcal{M}_{n\times n}(\mathbb{C}): Z^t=Z, Y>0\}$, where $Z=X+iY$, be the usual Siegel upper half plane. We define $j(\gamma,Z)=\det(C_\gamma Z+D_\gamma)$.

The non-holomorphic Eisenstein series $E_n^{k,s}$ is defined as $$E_n^{k,s}(Z)=\det(Y)^s\sum_{\Gamma_n^0\backslash\Gamma_n}j(\gamma,Z)^{-k}|j(\gamma,Z)|^{-2s}$$ for $k\in\mathbb{N}$ and $s\in\mathbb{C}$. In order to prove its absolute convergence, we need to estimate $$\sum_{\Gamma_n^0\backslash\Gamma_n}|j(\gamma,Z)|^{\alpha}$$ where $\alpha\in\mathbb{R}$. I know that there are explicit conditions on $\alpha(k,s,n)$ so that the series is convergent, but I could not find any reference for a proof.

If $n=1$, this series turns out to be $\le c_0\sum_{c,d\in\mathbb{Z}}^{(c,d)\not=(0,0)}|cz+d|^{\alpha}$ for some constant $c_0$, and I have seen (again no reference) that this is $\le c_1 y^{\alpha+1}$ on the usual fundamental domain for $\Gamma_1\backslash \mathcal{H}_1$ (which I am happy to believe, as it looks like taking that sum is more or less like integrating over one variable, therefore raising the exponent by one).

Is something similar happening in general $n\ge 1$, maybe like $$\sum_{\Gamma_n^0\backslash\Gamma_n}|j(\gamma,Z)|^{\alpha}\le c_0 \det(Y)^{\alpha+n}$$ on a fundamental domain for $\Gamma_n\backslash \mathcal{H}_n$ chosen so that $Y\ge c_1 I$ (like Klingen explains, using the Minkowski reduced set)? Can you suggest any reference on the topic?


1 Answer 1


Borel's article in the Boulder Conference (Proc Symp AMS IX) gives a quite general method for proof of convergence of Eisenstein series, which he attributes to Godement.

Asymptotics for Eisenstein series, even when meromorphically continued are given by the "theory of the constant term", which asserts that the constant term(s) dominate the asymptotics of a moderate-growth, K-finite, $\mathfrak z$-finite automorphic form. If I recall correctly, proof of this is either in Borel's article in IX, or is in one of the two Godement articles there.

  • $\begingroup$ I cannot find anywhere the claim about the constant term, possibly because I am unfamiliar with the language of those articles. Do you know where else I could find information about it? I could not find anything online by searching "theory of the constant term".. $\endgroup$ Oct 16, 2018 at 0:32

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