Level Reciprocity in the twisted second moment of Rankin-Selberg L-functions

TL;DR

This paper establishes an exact reciprocity formula for the twisted second moment of Rankin-Selberg L-functions, revealing a deep symmetry between the twist parameter and the level in automorphic forms.

Contribution

It introduces a novel reciprocity relation for the second moment of Rankin-Selberg L-functions, using trace formulas and Kloosterman sum identities.

Findings

Proves an exact formula for the twisted second moment of L-functions.

Establishes a reciprocity relation exchanging twist parameter and level.

Employs trace formulas and Kloosterman sum identities to derive the result.

Abstract

We prove an exact formula for the second moment of Rankin-Selberg $L$-functions $L(1/2,f \times g)$ twisted by $\lambda_f(p)$, where $g$ is a fixed holomorphic cusp form and $f$ is summed over automorphic forms of a given level $q$. The formula is a reciprocity relation that exchanges the twist parameter $p$ and the level $q$. The method involves the Bruggeman/Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.