Average of Central L-values for GL(2)$\times$GL(1), Hybrid Subconvexity, and Simultaneous Nonvanishing

TL;DR

This paper develops a new method using a regularized relative trace formula to estimate second moments of twisted L-functions over number fields, achieving hybrid subconvex bounds and addressing simultaneous nonvanishing.

Contribution

It introduces a novel application of the regularized relative trace formula to obtain hybrid subconvexity bounds and solve the simultaneous nonvanishing problem for L-functions.

Findings

Established second moment estimates for twisted L-functions.

Achieved hybrid subconvex bounds comparable to the Weyl bound.

Provided an application to simultaneous nonvanishing of L-values.

Abstract

We employ a regularized relative trace formula to establish a second moment estimate for twisted $L$-functions across all aspects over a number field. Our results yield hybrid subconvex bounds for both Hecke $L$-functions and twisted $L$-functions, comparable to the Weyl bound in suitable ranges. Moreover, we present an application of our results to address the simultaneous nonvanishing problem.