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

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.