Fourth Moment of Automorphic Forms

Abstract: 

The amplifying technique of Iwaniec and Sarnak has proven to be an influential tool when it comes to bounding sup-norms of automorphic forms. In their seminal work, they further mentioned how to improve their technique given an arithmetic input. This arithmetic input is available if one can deal with a long amplifier. Unfortunately, their technique does not allow for such a long amplifier. We devise a new strategy using a theta kernel on $SO_3 \times SO_3 \times SL_2$, which naturally encompasses such a maximal length amplifier with the additional benefit of being able to bound a fourth moment rather than an individual form. We apply this technique to holomorphic Hecke eigenforms with respect to lattices coming from indefinite quaternion algebras over $\mathbb Q$ in the weight aspect as well as the level aspect. This is joint work with Ilya Khayutin and Paul Nelson.