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

2025/11/18 by Darbar, Pranendu
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.14012

Abstract

We improve the range of uniformity in the double-exponential decay of the tail of the distribution established by Lumley~\citeLumley 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~\mathbbF_q, in the limit as g → ∞. 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.

Related

• Correlation of shifted values of L-functions in the hyperelliptic ensemble
• The fluctuations in the number of points on a hyperelliptic curve over a finite field
• Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and Random Matrix Theory
• Large values of quadratic Dirichlet L-functions
• Improving the error term in the mean value of L(\tfrac12,χ) in the hyperelliptic ensemble