Central Values of $L$-Functions of Twisted Modular Forms and Local Polynomials

TL;DR

This paper demonstrates that the vanishing of the product of two central $L$-values of twisted modular forms can be determined by evaluating a local polynomial at finitely many points, using harmonic Maass forms and Shimura-Shintani correspondence.

Contribution

It introduces a method to decide the vanishing of $L$-function products via local polynomial evaluations, extending previous results with new connections to harmonic Maass forms.

Findings

Local polynomial evaluation suffices to determine zero product of $L$-values.

Relates local polynomial to $L$-function products through harmonic Maass forms.

Extends prior results by Ehlen et al. and Males et al.

Abstract

In this paper we study the product of two central values of $L$-functions of a twisted modular. We show that it suffices to compute a local polynomial at a finite number of points to decide whether the product is zero. For the proof, we relate the local polynomial to the product of the $L$-functions using a locally harmonic Maass form and building on the Shimura-Shintani correspondence. This extends results from Ehlen, Guerzhoy, Kane and Rolen as well as Males, Mono, Rolen and Wagner.