Adelic Voronoi Summation and Subconvexity for GL(2) L-Functions in the depth Aspect

TL;DR

This paper develops a new explicit Voronoi summation formula and applies it to achieve subconvexity bounds for degree two automorphic L-functions in the depth aspect, focusing on prime power conductor twists.

Contribution

It introduces a flexible Voronoi summation formula and uses it to establish subconvexity bounds in the depth aspect for GL(2) L-functions, extending p-adic analogues of known results.

Findings

Established a new explicit Voronoi summation formula.

Proved subconvexity bounds close to Weyl strength in the depth aspect.

Extended the understanding of L-functions in the p-adic setting.

Abstract

In this paper we establish a very flexible and explicit Voronoi summation formula. This is then used to prove an almost Weyl strength subconvexity result for automorphic $L$-functions of degree two in the depth aspect. That is, looking at twists by characters of prime power conductor. This is the natural $p$-adic analogue to the well studied $t$-aspect.