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

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 $n=1$.