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

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.