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

Abstract

Let $F$ be a number field with adele ring ${\mathbb{A}}_F$, ${\pi }_1,{\pi }_2$ 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 $(\mathfrak{ql},\mathfrak{uv})=1$. 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 $\mathfrak{uvq}$. 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 $\mathfrak{q}$.

16 pages. This is a preliminary draft. Comments are welcome! arXiv admin note: substantial text overlap with arXiv:2501.04022; substantial text overlap with arXiv:1912.01512, arXiv:1809.10593 by other authors