Yoshida lifts and simultaneous non-vanishing of dihedral twists of modular L-functions

TL;DR

This paper proves that for certain elliptic modular forms, there are infinitely many imaginary quadratic fields and characters where their twisted L-functions do not vanish at the critical point, using Yoshida liftings and Fourier coefficient non-vanishing.

Contribution

It establishes a quantitative non-vanishing result for twisted L-functions of modular forms via Yoshida liftings and Fourier coefficient analysis.

Findings

Infinitely many imaginary quadratic fields with non-vanishing L-values.

Non-vanishing results for Fourier coefficients of Siegel modular forms.

Application of Yoshida liftings to non-vanishing problems.

Abstract

Given elliptic modular forms f and g satisfying certain conditions on their weights and levels, we prove (a quantitative version of the statement) that there exist infinitely many imaginary quadratic fields K and characters chi of the ideal class group Cl_K such that L(1/2, f_K \times chi) \neq 0 and L(1/2, g_K \times chi) \neq 0. The proof is based on a non-vanishing result for Fourier coefficients of Siegel modular forms combined with the theory of Yoshida liftings.