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

Abstract

Let $F$ be a number field with adele ring $\mathbb{A}_F$, $π_1, π_2$ 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 $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$ twisted by the Hecke eigenvalues $λ_π(\mathfrak{l})$, where $π$ 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 $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$ in terms of the norm of $\mathfrak{q}$.