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

TL;DR

The paper develops a spectral reciprocity framework for the twisted first moment of triple product $L$-functions, relating $L(1/2, π ⊗ π_1 ⊗ π_2)$ twisted by Hecke data to a spectral expansion of triple-product periods. By leveraging integral representations (via the Ichino formula), spectral decomposition, and Plancherel theory, it links global central values to local period data and constructs a symmetric reciprocity between conductor ranges. The main achievement is a subconvexity bound in the level aspect for $L(1/2, π_1 ⊗ π_2 ⊗ π_3)$, obtained through an amplifier and a careful analysis of the symmetric period, yielding a bound of the form $L(1/2, π_1 ⊗ π_2 ⊗ π_3) ≪ q^{1-δ+ε}$ with $δ=(1/2-θ)(1-2θ_1-2θ_2)/(3-2θ_1-2θ_2)$; under Ramanujan, the exponent improves to $δ=1/6$. The approach highlights the role of the archimedean factor $f(π_∞)$ and provides a robust framework for subconvexity in GL$_2 imes$GL$_2 imes$GL$_2$ in the finite level aspect, with potential applicability to related automorphic $L$-functions. Overall, the work advances subconvexity theory by blending spectral reciprocity with amplification and explicit period analysis.

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 $\mathfrak{q}$.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 3.1
• Remark 4.1
• Proposition 4.2
• proof
• Proposition 4.3
• proof