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Shifted moments of modular $L$-functions to a fixed level

Peng Gao, Liangyi Zhao

TL;DR

This paper studies upper bounds for shifted moments of modular $L$-functions at a fixed prime level $N$, under GRH. Focusing on the family of holomorphic cusp forms $H^*_ appa(N)$, it bounds averages of products of central values $|L(\tfrac12+i t_j,f)|^{a_j}$ with $|t_j|\le N^A$, revealing the orthogonal symmetry type of the family. The approach combines a Dirichlet-polynomial approximation to $\log|L|$ with a Harper-style dyadic prime-decomposition and a careful control of main and exceptional contributions via sets $\mathcal{S}(j)$ and $\mathcal{P}(m)$, ultimately establishing an upper bound of the form $\sum_{f\in H^*_{\kappa}(N)} \prod_{j=1}^k |L(\tfrac12+i t_j,f)|^{a_j} \ll N$. These results extend the shifted-moment framework to orthogonal families and provide evidence for the expected symmetry type, complementing previous unitary and symplectic bounds in the literature.

Abstract

We establish upper bounds for shifted moments of modular $L$-functions to a fixed prime level under the generalized Riemann hypothesis.

Shifted moments of modular $L$-functions to a fixed level

TL;DR

This paper studies upper bounds for shifted moments of modular -functions at a fixed prime level , under GRH. Focusing on the family of holomorphic cusp forms , it bounds averages of products of central values with , revealing the orthogonal symmetry type of the family. The approach combines a Dirichlet-polynomial approximation to with a Harper-style dyadic prime-decomposition and a careful control of main and exceptional contributions via sets and , ultimately establishing an upper bound of the form . These results extend the shifted-moment framework to orthogonal families and provide evidence for the expected symmetry type, complementing previous unitary and symplectic bounds in the literature.

Abstract

We establish upper bounds for shifted moments of modular -functions to a fixed prime level under the generalized Riemann hypothesis.
Paper Structure (6 sections, 8 theorems, 61 equations)

This paper contains 6 sections, 8 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cusp form $L$-functions
  4. Sums over primes
  5. Upper bounds for modular $L$-functions
  6. Proof of Theorem \ref{['t1']}

Key Result

Theorem 1.1

With the notation as above and the truth of GRH, let $k\geq 1$ be a fixed integer and $a_1,\ldots, a_{k}$, $A$ fixed positive real numbers. Suppose that $N$ is a large prime number and $t=(t_1,\ldots ,t_{k})$ a real $k$-tuple with $|t_j|\leq N^A$. Then The implied constant depends on $k$, $A$ and the $a_j$'s, but not on $N$ or the $t_j$'s.

Theorems & Definitions (10)

  • Theorem 1.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.6
  • proof
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • proof