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

Richard Griffon

doi:10.5802/jtnb.1065

Explicit

L

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

Richard Griffon ¹

¹ Universiteit Leiden – Mathematisch Instituut Postbus 9512 2300 RA Leiden, The Netherlands

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 1059-1084.

Résumé
Abstract

Étant donné un corps fini $𝔽_{q}$ de caractéristique $p \geq 5$ , nous considérons la famille de courbes elliptiques $E_{d}$ définies sur $K = 𝔽_{q} (t)$ par $E_{d} : y^{2} + x y - t^{d} y = x^{3}$ , pour tout entier $d \geq 1$ qui est premier à $q$ .

Nous donnons une expression explicite des fonctions $L$ de ces courbes. De plus, nous déduisons de ce calcul que les courbes $E_{d}$ satisfont un analogue du théorème de Brauer–Siegel. Plus spécifiquement, nous montrons que, lorsque $d \to \infty$ parcourt les entiers premiers à $q$ , l’on a

log (| Ш (E_{d} / K) | \cdot Reg (E_{d} / K)) \sim log H (E_{d} / K),

où $H (E_{d} / K)$ désigne la hauteur différentielle exponentielle de $E_{d}$ , $Ш (E_{d} / K)$ son groupe de Tate–Shafarevich et $Reg (E_{d} / K)$ son régulateur de Néron–Tate.

For a finite field $𝔽_{q}$ of characteristic $p \geq 5$ and $K = 𝔽_{q} (t)$ , we consider the family of elliptic curves $E_{d}$ over $K$ given by $y^{2} + x y - t^{d} y = x^{3}$ 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 $d \to \infty$ ranging over the integers coprime with $q$ , one has

log (| Ш (E_{d} / K) | \cdot Reg (E_{d} / K)) \sim log H (E_{d} / K),

where $H (E_{d} / K)$ denotes the exponential differential height of $E_{d}$ , $Ш (E_{d} / K)$ its Tate–Shafarevich group and $Reg (E_{d} / K)$ its Néron–Tate regulator.

Reçu le : 2017-12-10
Accepté le : 2018-06-05
Publié le : 2019-03-28

MR Zbl

DOI : 10.5802/jtnb.1065

Classification : 11G05, 11G40, 14G10, 11F67, 11M38
Mots clés : Elliptic curves over function fields, Explicit computation of $L$-functions, Special values of $L$-functions and BSD conjecture, Estimates of special values, Analogue of the Brauer–Siegel theorem.

Affiliations des auteurs :

Richard Griffon ¹

¹ Universiteit Leiden – Mathematisch Instituut Postbus 9512 2300 RA Leiden, The Netherlands

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Richard Griffon},
     title = {Explicit $L$-functions and a {Brauer{\textendash}Siegel} theorem for {Hessian} elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {1059--1084},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1065},
     zbl = {1441.11143},
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Richard Griffon. Explicit $L$-functions and a Brauer–Siegel theorem for Hessian elliptic curves. Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 1059-1084. doi : 10.5802/jtnb.1065. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1065/

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