On the Distribution and Maximal Behavior of $L(1, \chi_D)$ over Hyperelliptic Curves

TL;DR

This paper enhances understanding of the distribution and extreme values of quadratic Dirichlet L-functions over hyperelliptic curves, extending uniformity ranges and providing lower bounds using advanced probabilistic methods.

Contribution

It improves the uniformity range for the tail distribution of $L(1, \,\chi_D)$ and applies a resonator method to establish lower bounds up to conjectural levels.

Findings

Extended the uniformity range in the tail distribution decay.

Established a double-exponential decay lower bound for the distribution.

Applied a long resonator method to approach conjectural behavior.

Abstract

We improve the range of uniformity in the double-exponential decay of the tail of the distribution established by Lumley~\cite{Lumley} for the quadratic Dirichlet $L$-function $L(1, \chi_D)$ over the ensemble of hyperelliptic curves of genus~$g$ defined over a fixed finite field~$\mathbb{F}_q$, in the limit as $g \to \infty$. Furthermore, we apply a long resonator method to show that this range of uniformity may persist up to its conjectural level by establishing a double-exponential decay lower bound for the corresponding distribution function.