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

Abstract

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.

On the arithmetic of a family of superelliptic curves

“…We note that there are several sequences of elliptic curves for which similar behaviour has been described. See [HP16,Gri16,Gri18,Gri19,GU20] For a fixed pair (a, b), the genus g of C = C a,b,q is constant as q varies. Hence the term log r g / log H(J) is o(1) as q → ∞.…”

Elliptic curves with large Tate-Shafarevich groups over $\mathbb{F}_q(t)$

“…This expression can be derived from the definition (1.1)-(1.2) of the L-function, just as in [Gri18] (see Lemma 4.6 and the following paragraph there) or [GU19,§4]. Here, we have implicitly used the fact that E γ,a has additive reduction at the place ∞ (see Proposition 2.2) to ignore the local terms corresponding to this place.…”

“…In a previous work, the first-named author has proved that, as a → ∞, the sequence of measures {µ a } a≥1 converges weak- * to µ ∞ in a quantitative way (see Theorem 6.6 in [Gri18]). More specifically, we have Theorem 6.3.…”

“…We study the relevant aspects of the distribution of Θ γ,a in section 6. Namely, we first recall from [Gri18] an effective version of an asymptotic distribution statement, the qualitative form of which says that, asymptotically as a → ∞, the set Θ γ,a becomes equidistributed in [−1, 1] with respect to the measure 2 π √ 1 − x 2 dx. Next we prove a quantitative version of the fact that Θ γ,a "avoids" the points −1, 0 and 1, which are the poles of x → w(x) on [−1, 1].…”

Sur le théorème de Brauer--Siegel généralisé

“…C'est un résultat central de la théorie des nombres du XX e siècle, qui a trouvé de nombreuses applications à des domaines variés des mathématiques (par exemple aux empilements de sphères [RT90], aux exposants de groupes de classes [HM18], en géométrie anabélienne [Iva14], pour la conjecture d'André-Oort pour A g [Tsi18]) et de l'informatique (théorie des graphes [ST96], cryptosystèmes à clé publique [HM00]). Récemment, le théorème de Brauer-Siegel a également donné lieu à des énoncés en dimension supérieure : ainsi, on pourra consulter [HP16] et [Gri16,Gri18a,Gri19] pour un analogue dans certaines suites de courbes elliptiques. Mentionnons enfin que, faisant suite à la preuve dans [Gri18b] d'un théorème de Brauer-Siegel pour la suite des surfaces de Fermat sur un corps fini, Hindry [Hin19] a proposé une vaste généralisation conjecturale pour des suites de variétés sur les corps finis.…”