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

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.

25 pages. Comments are very welcome!

References in corpus (1)

• Analogue of the Brauer-Siegel theorem for Legendre elliptic curves

Cited by in corpus (2)

• On the arithmetic of a family of twisted constant elliptic curves
• Explicit L-functions and a Brauer-Siegel theorem for Hessian elliptic curves