Spectral Reciprocity and Hybrid Subconvexity Bound for triple product $L$-functions

Abstract

Let $F$ be a number field with adele ring ${\mathbb{A}}_F$, ${\pi }_1,{\pi }_2$ be two unitary cuspidal automorphic representations of ${\mathrm{PGL}}_2({\mathbb{A}}_F)$ with finite analytic conductor. We study the twisted first moment of the triple product $L$-function $L(\frac{1}{2},\pi \otimes {\pi }_1\otimes {\pi }_2)$ and the Hecke eigenvalues ${\lambda }_{\pi }(\mathfrak{l})$, where $\pi$ is a unitary automorphic representation of ${\mathrm{PGL}}_2({\mathbb{A}}_F)$ and $\mathfrak{l}$ is an integral ideal coprimes with the finite analytic conductor $C(\pi \otimes {\pi }_1\otimes {\pi }_2)$. The estimation becomes a reciprocity formula between different moments of $L$-functions. Combining with the ideas and estimations established in [HMN23] and [MV10], we study the subconvexity problem for the triple product $L$-function in the level aspect and give a new explicit hybrid subconvexity bound for $L(\frac{1}{2},\pi \otimes {\pi }_1\otimes {\pi }_2)$, allowing joint ramifications and conductor dropping range.

23 pages. This is a preliminary draft. Comments are welcome! arXiv admin note: substantial text overlap with arXiv:1912.01512 by other authors