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

Griffon, Richard

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

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

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

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

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 $E_d:y^2+xy-t^{d}y=x^3$, 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

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

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

Classification: 11G05, 11G40, 14G10, 11F67, 11M38
Keywords: 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.

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