Explicit L-functions and a Brauer–Siegel theorem for Hessian elliptic curves
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 1059-1084.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1065
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.
Griffon, Richard 1

1 Universiteit Leiden – Mathematisch Instituut Postbus 9512 2300 RA Leiden, The Netherlands
@article{JTNB_2018__30_3_1059_0,
     author = {Griffon, Richard},
     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},
     mrnumber = {3938642},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1065/}
}
TY  - JOUR
AU  - Griffon, Richard
TI  - Explicit $L$-functions and a Brauer–Siegel theorem for Hessian elliptic curves
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 1059
EP  - 1084
VL  - 30
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1065/
DO  - 10.5802/jtnb.1065
LA  - en
ID  - JTNB_2018__30_3_1059_0
ER  - 
%0 Journal Article
%A Griffon, Richard
%T Explicit $L$-functions and a Brauer–Siegel theorem for Hessian elliptic curves
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 1059-1084
%V 30
%N 3
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1065/
%R 10.5802/jtnb.1065
%G en
%F JTNB_2018__30_3_1059_0
Griffon, Richard. Explicit $L$-functions and a Brauer–Siegel theorem for Hessian elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 1059-1084. doi : 10.5802/jtnb.1065. http://www.numdam.org/articles/10.5802/jtnb.1065/

[1] Cohen, Henri Number theory. Vol. I. Tools and Diophantine equations, Graduate Texts in Mathematics, 239, Springer, 2007 | MR | Zbl

[2] Conceição, Ricardo P.; Hall, Chris; Ulmer, Douglas Explicit points on the Legendre curve II, Math. Res. Lett., Volume 21 (2014) no. 2, pp. 261-280 | DOI | MR | Zbl

[3] Davis, Christopher; Occhipinti, Tommy Explicit points on y 2 +xy-t d y=x 3 and related character sums, J. Number Theory, Volume 168 (2016), pp. 13-38 | DOI | MR | Zbl

[4] Griffon, Richard Analogues du théorème de Brauer–Siegel pour quelques familles de courbes elliptiques, Université Paris Diderot (France) (2016) (Ph. D. Thesis)

[5] Griffon, Richard Analogue of the Brauer–Siegel theorem for Legendre elliptic curves, J. Number Theory, Volume 193 (2018), pp. 189-212 | DOI | MR | Zbl

[6] Griffon, Richard Bounds on special values of L-functions of elliptic curves in an Artin-Schreier family (2018) (to appear in European J. Math) | Zbl

[7] Griffon, Richard A Brauer–Siegel theorem for Fermat surfaces over finite fields, J. Lond. Math. Soc., Volume 97 (2018) no. 3, pp. 523-549 | DOI | MR | Zbl

[8] Hardy, Godfrey H.; Wright, Edward M. An introduction to the theory of numbers, Oxford University Press, 2008 | Zbl

[9] Hindry, Marc Why is it difficult to compute the Mordell–Weil group?, Diophantine geometry (Centro di Ricerca Matematica Ennio De Giorgi (CRM)), Volume 4, Edizioni della Normale, 2007, pp. 197-219 | MR | Zbl

[10] Hindry, Marc; Pacheco, Amìlcar An analogue of the Brauer–Siegel theorem for abelian varieties in positive characteristic, Mosc. Math. J., Volume 16 (2016) no. 1, pp. 45-93 | DOI | MR | Zbl

[11] Lang, Serge Conjectured Diophantine estimates on elliptic curves, Arithmetic and geometry, Vol. I (Progress in Mathematics), Volume 35, Birkhäuser, 1983, pp. 155-171 | DOI | MR | Zbl

[12] Lang, Serge Algebraic number theory, Graduate Texts in Mathematics, 110, Springer, 1994

[13] Schütt, Matthias; Shioda, Tetsuji Elliptic surfaces, Algebraic geometry in East Asia—Seoul 2008 (Advanced Studies in Pure Mathematics), Volume 60, Mathematical Society of Japan, 2008, pp. 51-160 | Zbl

[14] Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, Springer, 1994 | MR | Zbl

[15] Silverman, Joseph H. The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, 2009 | MR | Zbl

[16] Tate, John T. On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9, Société Mathématique de France, 1965, pp. 415-440 | Zbl

[17] Ulmer, Douglas Elliptic curves with large rank over function fields, Ann. Math., Volume 155 (2002) no. 1, pp. 295-315 | DOI | MR | Zbl

[18] Ulmer, Douglas L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields, Invent. Math., Volume 167 (2007) no. 2, pp. 379-408 | DOI | MR | Zbl

[19] Ulmer, Douglas Elliptic curves over function fields, Arithmetic of L-functions (IAS/Park City Mathematics Series), Volume 18, American Mathematical Society, 2011, pp. 211-280 | DOI | MR | Zbl

[20] Ulmer, Douglas Explicit points on the Legendre curve, J. Number Theory, Volume 136 (2014), pp. 165-194 | DOI | MR | Zbl

Cited by Sources: