Low-lying zeros in families of holomorphic cusp forms: the weight aspect

TL;DR

This paper investigates the distribution of low-lying zeros of L-functions associated with holomorphic cusp forms of large weight, refining previous results by identifying lower-order terms and providing unconditional estimates.

Contribution

It refines the density results of low-lying zeros in the weight aspect, uncovers lower-order terms with a sharp transition at support 1, and offers unconditional, more precise estimates.

Findings

Identified lower-order terms involving f1f1f1f1(1) in the zero density.

Discovered a sharp transition in the zero distribution at support 1.

Provided unconditional estimates surpassing Ratios Conjecture predictions.

Abstract

We study low-lying zeros of $L$-functions attached to holomorphic cusp forms of level $1$ and large weight. In this family, the Katz--Sarnak heuristic with orthogonal symmetry type was established in the work of Iwaniec, Luo and Sarnak for test functions $\phi$ satisfying the condition supp$(\widehat \phi) \subset(-2,2)$. We refine their density result by uncovering lower-order terms that exhibit a sharp transition when the support of $\widehat \phi$ reaches the point $1$. In particular the first of these terms involves the quantity $\widehat \phi(1)$ which appeared in previous work of Fouvry--Iwaniec and Rudnick in symplectic families. Our approach involves a careful analysis of the Petersson formula and circumvents the assumption of GRH for $\text{GL}(2)$ automorphic $L$-functions. Finally, when supp$(\widehat \phi)\subset (-1,1)$ we obtain an unconditional estimate which is significantly more precise than the prediction of the $L$-functions Ratios Conjecture.