Averages and nonvanishing of central values of triple product $L$-functions

TL;DR

This paper derives exact average formulas for central values of triple product L-functions of cusp forms and applies these results to establish lower bounds on nonvanishing cases, advancing understanding of these special values.

Contribution

It introduces a novel combination of Ichino's period formula and a relative trace formula to compute precise averages of triple product L-values.

Findings

Exact averages of $L(3k-1,f\times g\times h)$ are obtained.

Lower bounds on the number of nonvanishing central L-values are established.

Applications to nonvanishing problems demonstrate the effectiveness of the average formulas.

Abstract

Let $f,g,h$ be three normalized cusp newforms of weight $2k$ on $\Gamma_0(N)$ which are eigenforms of Hecke operators. We use Ichino's period formula combined with a relative trace formula to show exact averages of $L(3k-1,f\times g\times h)$. We also present some applications of the average formulas to the nonvanishing problem, giving a lower bound on the number of nonvanishing central $L$-values when one of the forms is fixed.