Bessel Periods on $U(2,1) \times U(1,1)$, Relative Trace Formula and Non-Vanishing of Central $L$-values

TL;DR

This paper computes the asymptotics of Bessel period moments for certain automorphic representations on unitary groups, leading to non-vanishing results for complex central L-values using a relative trace formula approach.

Contribution

It introduces a novel application of the relative trace formula to evaluate orbital integrals and establish non-vanishing of high-degree central L-values in a challenging conductor dropping setting.

Findings

Asymptotic formulas for second moments of Bessel periods

Quantitative non-vanishing results for degree twelve L-values

Implications for the rank of associated Selmer groups

Abstract

In this paper we calculate the asymptotics of the second moment of the Bessel periods associated to certain holomorphic cuspidal representations $(\pi, \pi')$ of $U(2,1) \times U(1,1)$ of regular infinity type (averaged over $\pi$). Using these, we obtain quantitative non-vanishing results for the Rankin-Selberg central $L$-values $L(1/2, \pi \times \pi')$, which are of degree twelve over $\mathbb{Q}$, with concomitant difficulty in applying standard methods, especially since we are in a `conductor dropping' situation. We use the relative trace formula, and the orbital integrals are evaluated rather than compared with others. Besides their intrinsic interest, non-vanishing of these critical values also lead, by known results, to deducing certain associated Selmer groups have rank zero.