Spectral Reciprocity for the first moment of triple product $L$-functions and applications

TL;DR

This paper establishes a spectral reciprocity formula linking the first moment of triple product L-functions to spectral expansions, and applies it to derive subconvex bounds in the level aspect for these L-functions.

Contribution

It introduces a new spectral reciprocity formula for triple product L-functions and applies it to obtain subconvexity bounds in the level aspect.

Findings

Derived a reciprocity formula connecting twisted moments and spectral expansions.

Established subconvex bounds for triple product L-functions in the level aspect.

Applied spectral methods to advance understanding of automorphic L-functions.

Abstract

Let $F$ be a number field with adele ring $\mathbb{A}_F$, $\pi_1, \pi_2$ be two fixed unitary automorphic representations of $\mathrm{PGL}_2(\mathbb{A}_F)$ with finite coprime analytic conductor $\mathfrak{u}$ and $\mathfrak{v}$, $\mathfrak{q},\mathfrak{l}$ be two coprime integral ideals with $(\mathfrak{q} \mathfrak{l}, \mathfrak{u} \mathfrak{v})=1$. Following [Zac20], we estimate the first moment of $L(\frac{1}{2}, \pi \otimes \pi_1 \otimes \pi_2)$ twisted by the Hecke eigenvalues $\lambda_{\pi}(\mathfrak{l})$, where $\pi$ runs over unitary automorphic representations of finite conductor dividing $\mathfrak{u}\mathfrak{v}\mathfrak{q}$. By applying the triple product integrals, spectral decomposition and Plancherel formula, we get a reciprocity formula links the twisted first moment of triple product $L$-functions to the spectral expansion of certain triple product periods over automorphic representations of finite conductor dividing $\mathfrak{l}$. As application, we study the subconvexity problem for the triple product $L$-function in the level aspect and give a subconvex bound for $L(\frac{1}{2}, \pi \otimes \pi_1 \otimes \pi_2)$ in terms of the norm of $\mathfrak{q}$.