We study a certain Artin-Schreier family of elliptic curves over the function field K = F q (t). 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". In the paper 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.

Let ‫ކ‬ r be a finite field of characteristic p > 3. For any power q of p, consider the elliptic curve E = E q,r defined by y 2 = x 3 + t q − t over K = ‫ކ‬ r (t). We describe several arithmetic invariants of E such as the rank of its Mordell-Weil group E(K ), the size of its Néron-Tate regulator Reg(E), and the order of its Tate-Shafarevich group X(E) (which we prove is finite). These invariants have radically different behaviors depending on the congruence class of p modulo 6. For instance X(E) either has trivial p-part or is a p-group. On the other hand, we show that the product |X(E)| Reg(E) has size comparable to r q/6 as q → ∞, regardless of p (mod 6). Our approach relies on the BSD conjecture, an explicit expression for the L-function of E, and a geometric analysis of the Néron model of E.

We prove an analogue of the Brauer-Siegel theorem for Fermat surfaces over a finite field Fq. Namely, letting F d be the Fermat surface of degree d over Fq and pg(F d ) be its geometric genus, we show that, for d → ∞ ranging over the set of integers coprime with q, one hasHere, Br(F d ) denotes the Brauer group of F d and Reg(F d ) the absolute value of a Gram determinant of the Néron-Severi group NS(F d ) with respect to the intersection form.

We prove an analogue of the Brauer-Siegel theorem for the Legendre elliptic curves over F q (t). Namely, denoting by E d the elliptic curve with model y 2 = x(x + 1)(x + t d ) over K = F q (t), we show that, for d → ∞ ranging over the integers coprime with q, one hasHere, H(E d /K) denotes the exponential differential height of E d , X(E d /K) its Tate-Shafarevich group (which is known to be finite), and Reg(E d /K) its Néron-Tate regulator.

For a finite field F q of characteristic p ≥ 5 and K = F q (t), we consider the family of elliptic curves E d over K given by y 2 + xy − t d y = x 3 for all integers d coprime to q. We provide an explicit expression for the L-functions of these curves. Moreover, we deduce from this calculation that the curves E d satisfy an analogue of the Brauer-Siegel theorem. Precisely, we show that, for d → ∞ ranging over the integers coprime with q, one haswhere H(E d /K) denotes the exponential differential height of E d , X(E d /K) its Tate-Shafarevich group and Reg(E d /K) its Néron-Tate regulator. Keywords: Elliptic curves over function fields, Explicit computation of L-functions, Special values of Lfunctions and BSD conjecture, Estimates of special values, Analogue of the Brauer-Siegel theorem.