Bounding Vanishing at the Central Point of Cuspidal Newforms

TL;DR

This paper extends the analysis of zeros of $L$-functions at the central point, providing improved bounds for cuspidal newforms by generalizing moment formulas and optimizing test functions, with significant computational advancements.

Contribution

It generalizes the $n$th centered moment formulas for zeros of $L$-functions to arbitrary test functions and achieves new record bounds through computational improvements and test function optimization.

Findings

Improved bounds on the order of vanishing for cuspidal newforms.

Significant computational advancements leading to world record bounds.

Enhanced bounds grow rapidly with the rank of the forms.

Abstract

The Katz-Sarnak Density Conjecture states that zeros of families of $L$-functions are well-modeled by eigenvalues of random matrix ensembles. For suitably restricted test functions, this correspondence yields upper bounds for the families' order of vanishing at the central point. We generalize previous results on the $n$\textsuperscript{th} centered moment of the distribution of zeros to allow arbitrary test functions. On the computational side, we use our improved formulas to obtain significantly better bounds on the order of vanishing for cuspidal newforms, setting world records for the quality of the bounds. We also discover better test functions that further optimize our bounds. We see improvement as early as the $5$\textsuperscript{th} order, and our bounds improve rapidly as the rank grows (more than one order of magnitude better for rank 10 and more than four orders of magnitude for rank 50).