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arXiv · 2607.06022

Correlations of zeros of a family of $L$-functions in function fields with symplectic symmetry

Julio Andrade·Asmaa Shamesaldeen
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Abstract

In this paper, we adapt the framework developed by Mason and Snaith to investigate the $n$-level density of zeros in the context of function fields. Specifically, we derive explicit formulas for the $n$-level density of zeros in families of quadratic Dirichlet $L$-functions associated with hyperelliptic curves of genus $g$ over the finite field $\mathbb{F}_{q}$. Employing Mason and Snaith's method, we obtain precise expressions for the $1$-level density in these families and extend the approach to higher-level densities. Furthermore, we apply the method to derive formulas for the $n$-level density of zeros in families of $L$-functions associated with prime characters. Our results are consistent with the findings of Andrade, Jung, and Shamesaldeen in the case $n=1$.

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Julio Andrade, Asmaa Shamesaldeen. 2026-07-07. Correlations of zeros of a family of $L$-functions in function fields with symplectic symmetry. https://arxiv.org/abs/2607.06022

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