The Petersson/Kuznetsov trace formula with prescribed local ramifications

TL;DR

This paper develops refined trace formulas with specific local ramification conditions, enabling improved analysis of L-functions and establishing a wider Weyl bound in a hybrid setting.

Contribution

It introduces shorter, more targeted trace formulas with prescribed local ramifications, advancing the study of Rankin-Selberg L-functions and subconvexity bounds.

Findings

Derived refined trace formulas with local ramification constraints

Achieved a wider Weyl bound in a hybrid setting for L-functions

Enhanced understanding of the spectral side for specific newforms

Abstract

In this paper we derive refined Petersson/Kuznetsov trace formulae with prescribed local ramifications. The spectral side of these formulae picks out newforms whose associated local components come from specific sub-families of representations of given level, and are much shorter compared with the classical versions. We use them to study the first moment and the subconvexity bound of certain Rankin-Selberg L-function in a hybrid setting, obtaining Weyl bound in a wider range compared to previous works.