Explicit L-functions and a Brauer–Siegel theorem for Hessian elliptic curves

Abstract

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.

On the Northcott property for special values of L-functions

“…as shown in [51,Lemma 3.7]. Recent works of Hindry [51], Hindry-Pacheco [52] and Griffon [39,40,41,42] on the analogue of the Brauer-Siegel estimate for abelian varieties show that the numerator of ( 16) is also expected to be comparable (in some cases) to H(A). Hence it is necessary to gain further evidence in order to be able to decide if a Northcott property for the special value L * (A, 1) associated to abelian varieties holds in some cases.…”

On the Northcott property for special values of L-functions

“…First of all, if one denotes by H.A/ the exponential of the stable Faltings height of A, one has that H.A/ 1 A H.A/.log.H.A/// dim.A/=2 ; as shown in Lemma 3.7 of [73]. Recent works of Hindry [73], Hindry-Pacheco [74] and Griffon [60][61][62][63] on the analogue of the Brauer-Siegel estimate for abelian varieties show that the numerator of (6.13) is also expected to be comparable (in some cases) to H.A/. Hence it is necessary to gain further evidence in order to be able to decide if a Northcott property for the special value L .A; 1/ associated to abelian varieties holds in some cases.…”