Zeros of quadratic Dirichlet $L$-functions in the hyperelliptic ensemble

TL;DR

This paper analyzes the zeros of quadratic Dirichlet L-functions over function fields, revealing lower order terms in their statistical distributions that challenge existing conjectures and inform non-vanishing results.

Contribution

It computes lower order terms in the 1-level density and pair correlation of zeros, providing new insights beyond the predictions of the Ratios Conjecture.

Findings

Identifies secondary and lower order terms in the 1-level density for specific Fourier support ranges.

Detects lower order terms in pair correlation under certain restrictions.

Derives non-vanishing results and bounds on simple zeros of quadratic Dirichlet L-functions.

Abstract

We study the $1$-level density and the pair correlation of zeros of quadratic Dirichlet $L$-functions in function fields, as we average over the ensemble $\mathcal{H}_{2g+1}$ of monic, square-free polynomials with coefficients in $\mathbb{F}_q[x]$. In the case of the $1$-level density, when the Fourier transform of the test function is supported in the restricted interval $(\frac{1}{3},1)$, we compute a secondary term of size $q^{-\frac{4g}{3}}/g$, which is not predicted by the Ratios Conjecture. Moreover, when the support is even more restricted, we obtain several lower order terms. For example, if the Fourier transform is supported in $(\frac{1}{3}, \frac{1}{2})$, we identify another lower order term of size $q^{-\frac{8g}{5}}/g$. We also compute the pair correlation, and as for the $1$-level density, we detect lower order terms under certain restrictions; for example, we see a term of size $q^{-g}/g^2$ when the Fourier transform is supported in $(\frac{1}{4},\frac{1}{2})$. The $1$-level density and the pair correlation allow us to obtain non-vanishing results for $L(\frac12,\chi_D)$, as well as lower bounds for the proportion of simple zeros of this family of $L$-functions.