Integral representation and critical $L$-values for holomorphic forms on $GS{p}_{2n}\times G{L}_1$

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.

This paper has been superseded by arXiv:1803.06227 and will no longer be updated

Cited by in corpus (2)

• On the distribution of Satake parameters for Siegel modular forms
• The special values of the standard $L$-functions for ${\mathrm{GSp}}_{2n}\times {\mathrm{GL}}_1$