The Second Moment of $\mathrm{GL}_4 \times \mathrm{GL}_2$ $L$-functions at Special Points

TL;DR

This work establishes a short-interval second-moment bound for $L(s_j,\phi \times u_j)$ with $\phi$ a fixed ${\rm SL}_4(\mathbb{Z})$ cusp form and $u_j$ varying over ${\rm SL}_2({\mathbb Z})$ Maass cusp forms. The authors provide a direct, Poisson-free proof by combining a twisted large sieve on short intervals, the Voronoï summation formula for ${\rm SL}_4$, and careful analysis of Hankel-type transforms, avoiding Eisenstein–Kloosterman cancellation and double Poisson. The resulting bound, for $T^{\varepsilon} \le M \le T^{1-\varepsilon}$, yields $\sum_{T<t_j\le T+M}|L(s_j,\phi\times u_j)|^2 \ll_{\varepsilon,\phi} T^{5+\varepsilon}/M^{3}$, which is optimal when $M \asymp T^{1-\varepsilon}$. The approach sharpens our understanding of short-interval moments for Rankin–Selberg $L$-values and demonstrates a powerful Poisson-free method via a twisted spectral large sieve.

Abstract

In this paper, we provide an alternative proof of Chandee and Li's result on the second moment of $\mathrm{GL}_4 \times \mathrm{GL}_2$ special $L$-values. Our method is conceptually more direct as it neither detects the `Eisenstein--Kloosterman' cancellation nor uses the Poisson summation.

Theorems & Definitions (20)

• Theorem 1.1
• Definition 2.1
• Lemma 2.1
• Lemma 3.1
• Lemma 3.2
• Theorem 3.1
• Remark 3.1
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3