1
$\begingroup$

Let $\mathbb{H}_g = \{ \tau \in GL_g(\mathbb{C}) | \; {^t\tau} = \tau, Im(\tau) >0\}$ be the Siegel upper half space. There are Eisenstein series $$ E_{2k}(\tau) := \sum_{\gamma\in (P_0\cap \Gamma)\backslash \Gamma}{1\big|_{2k}\gamma} = \sum_{\gamma\in (P_0\cap \Gamma)\backslash \Gamma}{det(C_\gamma \tau + D_\gamma)^{-2k}}$$ on $\mathbb{H}_g$ (here $P_0$ is the subgroup of $\Gamma:=Sp_{2g}(\mathbb{Z})$ consisting of those matrices whose lower left $g\times g$-block is the zero matrix). Consider the following normalization $$ G_{2k}(\tau):= \frac{(2k-1)!\zeta(2k)}{(2\pi i)^{2k}}\cdot E_{2k}(\tau).$$ For $g=1$ one knows that $G_2(\tau)$ transforms as follows w.r.t. an element $\begin{pmatrix} a & b \\ c & d \end{pmatrix} =:\gamma \in SL_2(\mathbb{Z})$ $$ G_2(\frac{a\tau + b}{c\tau +d}) (c\tau +d)^{-2} = G_2(\tau) - \frac{c}{4\pi i(c\tau + d)}.$$ Does anybody know how this transformation looks like for $g>1$ w.r.t. $\, \gamma \in Sp_{2g}(\mathbb{Z})$ and/or indicate literature on this?

I'm only interested in classical (meaning scalar-valued) Siegel modular forms.

$\endgroup$
1
$\begingroup$

You haven't defined any of the notation in your question so it's difficult to give a specific answer, but there is certainly literature on this. These weight 2 Eisenstein series are examples of nearly holomorphic modular forms, and Shimura has defined nearly holomorphic automorphic forms on symplectic or unitary groups; see his paper "On a class of nearly holomorphic automorphic forms" (Annals of Math, 1986).

$\endgroup$
2
  • $\begingroup$ You're right...I've updated my question. $\endgroup$ – Sebastian Thyssen Jul 1 '14 at 15:53
  • $\begingroup$ Actually, I didn't find the answer to my question in this paper...do you know it's in there or was that just a guess? $\endgroup$ – Sebastian Thyssen Jul 1 '14 at 20:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.