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

TL;DR

The paper studies the tail distribution and maximal size of $L(1, \chi_D)$ as $D$ ranges over the hyperelliptic ensemble in function fields, establishing an extended uniformity range for the tail decay. It combines two core approaches: (i) computing high moments of a truncated Euler product and applying a saddle-point analysis to obtener a double-exponential decay bound, and (ii) the long resonator method to push lower bounds toward the conjectured maximal scale, aided by refined character-sum estimates. The main results show that the tail decays doubly exponentially up to a nearly conjectural regime, with explicit constants and an unconditional bound that approaches the heuristic picture; moreover, the resonance method yields a positive shift $C_2(q)>0$ for large $q$, refining the maximal order of $L(1, \chi_D)$. These findings connect the distribution of $L$-values to class-number-type invariants $h_D$ and regulators, offering evidence toward the expected maximal order $\log n + \log_2 n + O(1)$ and providing tools potentially adaptable to broader $GL(1)$ families in function fields.

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 $L(1, χ_D)$ 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.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['main theorem']}
• proof : Proof of Theorem \ref{['theorem on resonance method']}