Zero density estimate for modular form $L$-functions in weight aspect

TL;DR

This paper establishes a zero density estimate for modular form $L$-functions in the weight aspect, providing insights into the distribution of zeros near the central point as the weight increases.

Contribution

It introduces a zero density estimate for $L$-functions of weight $k$ cusp forms, advancing understanding of their zeros in the weight aspect.

Findings

Zero density estimate near the central point as weight $k$ grows

Unconditional upper bounds on distribution of central values

Supports further analysis of $L$-function zeros and values

Abstract

Considering the family of $L$-functions $\{L(s,f)\}_{f \in H_k}$ where $H_k$ is the set of weight $k$ Hecke-eigen cusp forms for $SL_2(\mathbb{Z})$, we prove a zero density estimate near the central point, valid as the weight $k \to \infty$. This is an ingredient in the author's related paper, which gives an unconditional upper bound on the distribution of the central values.