 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.

Étant donné un corps fini ${𝔽}_{q}$ de caractéristique $p\ge 5$, nous considérons la famille de courbes elliptiques ${E}_{d}$ définies sur $K={𝔽}_{q}\left(t\right)$ par ${E}_{d}:\phantom{\rule{4pt}{0ex}}{y}^{2}+xy-{t}^{d}y={x}^{3}$, pour tout entier $d\ge 1$ 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

$log\left(|Ш\left({E}_{d}/K\right)|·\mathrm{Reg}\left({E}_{d}/K\right)\right)\sim logH\left({E}_{d}/K\right),$

$H\left({E}_{d}/K\right)$ désigne la hauteur différentielle exponentielle de ${E}_{d}$, $Ш\left({E}_{d}/K\right)$ son groupe de Tate–Shafarevich et $\mathrm{Reg}\left({E}_{d}/K\right)$ son régulateur de Néron–Tate.

For a finite field ${𝔽}_{q}$ of characteristic $p\ge 5$ and $K={𝔽}_{q}\left(t\right)$, 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 $d\to \infty$ ranging over the integers coprime with $q$, one has

$log\left(|Ш\left({E}_{d}/K\right)|·\mathrm{Reg}\left({E}_{d}/K\right)\right)\sim logH\left({E}_{d}/K\right),$

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

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DOI : https://doi.org/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.
@article{JTNB_2018__30_3_1059_0,
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},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1065/}
}
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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