Linear Relations of Siegel Poincare Series and Nonvanshing of the Central Value of Spinor L-functions

TL;DR

This paper explores linear relations among Siegel Poincaré series and applies these findings to demonstrate the non-vanishing of Fourier coefficients and central L-values of Siegel cusp forms.

Contribution

It establishes new linear relations among Siegel Poincaré series and uses them to prove non-vanishing results for Fourier coefficients and central L-values.

Findings

Linear relations among a family of Siegel Poincaré series

Non-vanishing of Fourier coefficients of Siegel cusp forms

Non-vanishing of central values of spinor L-functions

Abstract

In this paper, we will first investigate the linear relations of a one parameter family of Siegel Poincar\'e series. Then we give the applications to the non-vanishing of Fourier coefficients of Siegel cusp eigenforms and the central values.