Xiaoyu Zhang: Non-vanishing of certain theta lifts modulo a prime (June 18, 2020)

Weil representations and theta lifts are important tools in transferring automorphic forms on one (classical) group to automorphic forms on another (classical) group. Rallis’ inner product formula further relates them to the study of special $L$-values and periods (Petersson products) of automorphic representations. To study the mod $p$ properties of special $L$-values and periods (the $p$-part of Bloch-Kato conjecture) through theta lifts, one needs to show that theta lift takes $p$-integral primitive automorphic forms to $p$-integral primitive ones after a suitable normalization. In this talk, I will try to introduce these ideas and discuss some results in this direction for the pair $(O(2n), Sp(2n))$ with $O(2n)$ compact. These results are based on some computations of Bessel periods and applications of Ratner’s theorems.