The fourth moment of holomorphic Hecke cusp forms in shorter intervals

TL;DR

This work establishes a power-saving asymptotic for the fourth moment of holomorphic Hecke cusp forms in a short weight interval for SL${}_2(\mathbb{Z})$. Central to the approach is Watson's formula, which links the fourth moment to central values of $L$-functions $L(\tfrac12,g)$ and $L(\tfrac12,\mathrm{sym}^2 f\times g)$, analyzed via approximate functional equations and the Petersson trace formula. The main term is shown to be $\frac{6}{\pi}$, while the off-diagonal contributions are controlled through delicate estimates of sums of Bessel functions and Kloosterman sums, with Poisson summation enabling a reduction to manageable ranges. The result holds for fixed $0<c\le\tfrac14$ and $H=K^{\frac{3}{4}+c}$, improving prior short-interval results and aligning with predictions from the random wave paradigm for holomorphic cusp forms.

Abstract

Let $0<c\le 1/4$ be fixed. For $H = K^{\frac{3}{4}+ c}$, we find the average value of the fourth moment of holomorphic Hecke cusp forms of weight varies within $[K,K+H]$, improving a previous result of Khan.

Theorems & Definitions (31)

• Conjecture 1.1: 1,Conjecture 1.2
• Theorem 1.2
• Remark 1.3
• Lemma 2.1: Petersson's trace formula
• proof
• Lemma 2.2: Approximate functional equation
• proof
• Lemma 2.3
• proof
• Lemma 3.1