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

2023/09/15 by Michel, Philippe, Ramakrishnan, Dinakar, Yang, Liyang · 1 citation
#11F67 #11F70 #11F72 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2309.08490

Abstract

In this paper we calculate the asymptotics of the second moment of the Bessel periods associated to certain holomorphic cuspidal representations (π, π') of U(2,1) × U(1,1) of regular infinity type (averaged over π). Using these, we obtain quantitative non-vanishing results for the Rankin-Selberg central L-values L(1/2, π× π'), which are of degree twelve over 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.

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