Pullbacks of Siegel Eisenstein Series and Weighted Averages of Critical $L$-values

TL;DR

This paper derives a weighted average formula for special values of L-functions associated with elliptic modular forms by analyzing Siegel Eisenstein series pullbacks and their spectral decomposition.

Contribution

It introduces a new method to compute weighted averages of critical L-values using pullbacks of Siegel Eisenstein series and explicit spectral analysis.

Findings

Derived a weighted average formula for L-values

Explicit spectral decomposition of Siegel Eisenstein series

Applicable to modular forms of weight k and full level

Abstract

In this paper we obtain a weighted average formula for special values of $L$-functions attached to normalized elliptic modular forms of weight $k$ and full level. These results are obtained by studying the pullback of a Siegel Eisenstein series and working out an explicit spectral decomposition.