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.


Raphael Steiner

Research Area

Number Theory Seminar


Institute for Advanced Study


Wed, 19/02/2020 - 2:00pm


RC-4082, The Red Centre, UNSW