The Mean Value of $L(\tfrac{1}{2},\chi)$ in the Hyperelliptic Ensemble

TL;DR

This paper derives an asymptotic formula for the average value of quadratic Dirichlet L-functions at the central point over hyperelliptic curves in function fields, aligning with predictions from Random Matrix Theory.

Contribution

It provides the first asymptotic formula for the first moment of quadratic L-functions over hyperelliptic ensembles in function fields, extending number field results to the function field setting.

Findings

Expected value of L(1/2,χ) computed for large genus g

Results agree with Random Matrix Theory conjectures

Analogous to Jutila's number-field results

Abstract

We obtain an asymptotic formula for the first moment of quadratic Dirichlet $L$--functions over function fields at the central point $s=\tfrac{1}{2}$. Specifically, we compute the expected value of $L(\tfrac{1}{2},\chi)$ for an ensemble of hyperelliptic curves of genus $g$ over a fixed finite field as $g\rightarrow\infty$. Our approach relies on the use of the analogue of the approximate functional equation for such $L$--functions. The results presented here are the function field analogues of those obtained previously by Jutila in the number-field setting and are consistent with recent general conjectures for the moments of $L$--functions motivated by Random Matrix Theory.