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

TL;DR

This work establishes a spectral reciprocity formula for the twisted first moment of the central triple product $L$-function on GL$_2$ over a number field, relating it to a second moment via a symmetric period expansion. By combining Ichino-Watson integral representations, sharp local bounds for period integrals, and archimedean analysis, the authors implement an amplification framework to obtain explicit subconvex bounds in the finite level (level) aspect for $L(\tfrac{1}{2},\pi_1\otimes\pi_2\otimes\pi_3)$. The main achievement is a quantitative hybrid subconvexity bound in terms of the analytic conductors $Q_f$ and $P_f$, namely $L(\tfrac{1}{2},\pi_1\otimes\pi_2\otimes\pi_3) \ll Q_f^{1/4+\varepsilon} P_f^{-(\tfrac{1}{4}-\tfrac{\theta}{2})(1-2\theta_1-2\theta_2)/(7-2\theta_1-2\theta_2)}$, with $0\le\theta_i\le 7/64$ and $\theta$ the best GL$(2)$-Ramanujan exponent. The approach also yields corollaries for Eisenstein inputs and for coprime squarefull conductors, and it highlights the interplay between global periods, local bounds, and spectral reciprocity in achieving subconvexity in higher rank settings. This provides a concrete, explicit framework for subconvexity via spectral reciprocity and amplification in the level aspect for triple product $L$-functions over number fields.

Abstract

Let $F$ be a number field with adele ring $\mathbb{A}_F$, $π_1, π_2$ 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 $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$ and the Hecke eigenvalues $λ_π(\mathfrak{l})$, where $π$ 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 $C(π\otimes π_1 \otimes π_2)$. 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 $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$, allowing joint ramifications and conductor dropping range.

Theorems & Definitions (20)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Proposition 3.1
• Remark 4.1
• Proposition 4.2