Theta functions, fourth moments of eigenforms, and the sup-norm problem I

Abstract

We give sharp point-wise bounds in the weight-aspect on fourth moments of modular forms on arithmetic hyperbolic surfaces associated to Eichler orders. Thereby, we strengthen a result of Xia and extend it to co-compact lattices. We realize this fourth moment by constructing a holomorphic theta kernel on G × G × SL_(2), for G an indefinite inner form of SL_2 over ℚ, based on the Bergman kernel, and considering its L^2 -norm in the Weil variable. The constructed theta kernel further gives rise to new elementary theta series for integral quadratic forms of signature (2,2).