Bounds on special values of L-functions of elliptic curves in an Artin\u2013Schreier family

Abstract

We study a certain Artin–Schreier family of elliptic curves over the function field . We prove an asymptotic estimate on the special values of their L-function in terms of the degree of their conductor; we show that the special values are, in a sense, ‘asymptotically as large as possible’. We also provide an explicit expression for their L-function. The proof of the main result uses this expression and a detailed study of the distribution of 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.