Integral representation and critical $L$-values for holomorphic forms on $GSp_{2n} \times GL_1$

TL;DR

This paper develops an explicit integral representation for twisted standard L-functions of holomorphic Siegel cusp forms, enabling the proof of a Deligne-predicted reciprocity law for critical values, even at arbitrary levels.

Contribution

It introduces a novel scalar-valued pullback formula for vector-valued Siegel cusp forms and computes archimedean integrals exactly, extending critical value results to arbitrary levels.

Findings

Explicit integral representation for twisted L-functions

Proof of a reciprocity law for critical L-values

Extension of results to arbitrary level cases

Abstract

We prove an explicit integral representation -- involving the pullback of a suitable Siegel Eisenstein series -- for the twisted standard $L$-function associated to a holomorphic vector-valued Siegel cusp form of degree $n$ and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to $L$-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case $n=2$, we are able to prove a reciprocity law -- predicted by Deligne's conjecture -- for the critical special values of the twisted standard $L$-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. The proof of this application uses our recent structure theorem [arXiv:1501.00524] for the space of nearly holomorphic Siegel modular forms of degree 2 and arbitrary level.