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.

• The paper establishes a spectral reciprocity formula linking twisted moments of triple product L-functions to related L-function families.
• It employs refined analytic techniques, including spectral decomposition, test vectors, and amplifiers, to tackle the subconvexity problem.
• The derived hybrid subconvexity bound advances level aspect estimates, thereby reducing computational complexity in evaluating central L-values.

Spectral Reciprocity and Hybrid Subconvexity Bound for Triple Product $L$-functions

The paper "Spectral Reciprocity and Hybrid Subconvexity Bound for Triple Product $L$-functions" (2501.04022) investigates the spectral reciprocity between moments of $L$-functions and derives new hybrid subconvexity bounds for triple product $L$-functions. This work specifically explores the twisted first moment of the triple product $L$-function $L(\frac{1}{2}, \pi \otimes \pi_1 \otimes \pi_2)$ concerning Hecke eigenvalues.

Introduction and Background

The paper addresses the challenge of subconvexity estimates, a pivotal problem in analytic number theory and the study of $L$-functions. The subconvexity problem seeks nontrivial upper bounds for $L(\frac{1}{2}, \Pi)$, expressed in the form $$L(\frac{1}{2}, \Pi) \ll_{F,\epsilon} C(\Pi)^{\frac{1}{4}-\delta+\epsilon}$$, where $C(\Pi)$ is the analytic conductor, and $\delta$ is a positive constant. Previous solutions to subconvexity have been broad, covering lower rank cases extensively while higher rank cases remain less understood, especially concerning hybrid aspects.

Spectral Reciprocity

A core contribution of the paper is the establishment of a spectral reciprocity formula between moments of different $L$-function families. By considering a unitary automorphic representation $\pi$ over $_2(_F)$, the study derives an identity involving local period integrals and Hecke eigenvalues that facilitates estimating twisted moments of triple $L$-functions. This reciprocity relates the sum of $L$-functions of one family to another, enabling the authors to leverage techniques from spectral decomposition and obtain new bounds.

Hybrid Subconvexity Bound

The hybrid subconvexity bound introduced is significant for triple product $L$-functions, particularly in the level aspect. The authors derive explicit bounds demonstrating that $L(\frac{1}{2}, \pi_1 \otimes \pi_2 \otimes \pi_3)$ is significantly bounded by $$Q_f^{1/4+\varepsilon} \cdot P_f^{-(\frac{1}{4}-\frac{\theta}{2})(1-2\theta_1-2\theta_2)/(7-2\theta_1-2\theta_2)}$$. Here, $Q_f$ and $P_f$ denote the products of conductors at the finite places, while $\theta$, $\theta_1$, and $\theta_2$ are parameters towards the Ramanujan-Petersson conjecture. The conditions assumed on the representations include coprimeness and particular constraints on analytic conductors, allowing for this sharp estimation.

Implementation Strategies

The theoretical results are framed within practical implementation strategies involving the choice of test vectors, amplifiers, and the use of non-Euclidean geometry to evaluate complex exponential integrals. Important computational steps involve Euler product formulations and symmetries in Hecke operators. Various lemmas and propositions provide detailed estimations of local integrals, which are crucial for implementation.

Conclusion

The paper bridges recent advances in both spectral theory and computational number theory by providing new insights into the behavior of triple product $L$-functions. The derived spectral reciprocity relates distinct $L$-function families, and the hybrid subconvexity bounds assert significant reductions in computational complexity for evaluating central $L$-values. Future work may extend these applications, targeting deeper spectral analyses and broader classes of automorphic forms.