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 p5, nous considérons la famille de courbes elliptiques E d définies sur K=𝔽 q (t) par E d :y 2 +xy-t d y=x 3 , pour tout entier d1 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 parcourt les entiers premiers à q, l’on a

log|Ш(Ed/K)|·Reg(Ed/K)logH(Ed/K),

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 p5 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 has

log|Ш(Ed/K)|·Reg(Ed/K)logH(Ed/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
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--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 = {jtnb.centre-mersenne.org/item/JTNB_2018__30_3_1059_0/}
}
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/item/JTNB_2018__30_3_1059_0/

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