Bounds on special values of L-functions of elliptic curves in an Artin-Schreier family

TL;DR

This paper investigates the asymptotic behavior of special values of L-functions for a family of elliptic curves over function fields, providing explicit formulas and connecting results to the Brauer--Siegel theorem via BSD conjecture.

Contribution

It establishes asymptotic estimates for L-function special values and derives explicit formulas for the family of elliptic curves studied.

Findings

Special values are asymptotically maximal relative to the conductor degree

Explicit expression for the L-function of the elliptic curves

Connection to the Brauer--Siegel theorem via BSD conjecture

Abstract

In this paper, we study a certain Artin--Schreier family of elliptic curves over the function field $\mathbb{F}_q(t)$. We prove an asymptotic estimate on the size of the special value of their $L$-function in terms of the degree of their conductor; loosely speaking, we show that the special values are "asymptotically as large as possible". We also provide an explicit expression for the $L$-function of the elliptic curves in the family. The proof of the main result uses this expression and a detailed study of the distribution of some character sums related to Kloosterman sums. Via the BSD conjecture, the main result translates into an analogue of the Brauer--Siegel theorem for these elliptic curves.