Sub-Weyl bound for $GL(2)$ $L$-functions

2:00pm, Wednesday 9th April

Abstract

We begin by briefly introducing the subconvexity problem for $L$-functions and the delta method, which has proven to be a powerful line of attack in this context. As applications, we obtain a sub-Weyl bound for $L$-functions associated to $SL(2,\mathbb{Z})$ forms, thereby crossing the Weyl barrier for the first time beyond $GL(1)$. We also derive a new bound for the Riemann zeta function that improves upon the previous record due to Bourgain. The proof uses a refinement of the `trivial' delta method.