On the cyclic torsion of elliptic curves over cubic number fields (II)
Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 663-670.

Le résultat de Merel sur la forme forte de la conjecture de borne uniforme a mis en valeur la classification des parties de torsion des groupes de Mordell–Weil des courbes elliptiques définies sur les corps de nombres de degré fixé d. Dans cet article, nous étudions les sous-groupes de torsion cycliques des courbes elliptiques sur les corps de nombres cubiques. Pour N=49,40,25 ou 22, nous montrons que /N n’est pas un sous-groupe de E(K) tor pour toute courbe elliptique E sur un corps de nombres cubique K.

Merel’s result on the strong uniform boundedness conjecture made it meaningful to classify the torsion part of the Mordell–Weil groups of all elliptic curves defined over number fields of fixed degree d. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For N=49,40,25 or 22, we show that /N is not a subgroup of E(K) tor for any elliptic curve E over a cubic number field K.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1100
Classification : 11G05, 11G18
Mots clés : torsion subgroup, elliptic curves, modular curves
Wang, Jian 1

1 College of Mathematics Jilin Normal University Siping, Jilin 136000, China
@article{JTNB_2019__31_3_663_0,
     author = {Wang, Jian},
     title = {On the cyclic torsion of elliptic curves over cubic number fields {(II)}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {663--670},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {3},
     year = {2019},
     doi = {10.5802/jtnb.1100},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1100/}
}
TY  - JOUR
AU  - Wang, Jian
TI  - On the cyclic torsion of elliptic curves over cubic number fields (II)
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2019
SP  - 663
EP  - 670
VL  - 31
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1100/
DO  - 10.5802/jtnb.1100
LA  - en
ID  - JTNB_2019__31_3_663_0
ER  - 
%0 Journal Article
%A Wang, Jian
%T On the cyclic torsion of elliptic curves over cubic number fields (II)
%J Journal de théorie des nombres de Bordeaux
%D 2019
%P 663-670
%V 31
%N 3
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1100/
%R 10.5802/jtnb.1100
%G en
%F JTNB_2019__31_3_663_0
Wang, Jian. On the cyclic torsion of elliptic curves over cubic number fields (II). Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 663-670. doi : 10.5802/jtnb.1100. http://www.numdam.org/articles/10.5802/jtnb.1100/

[1] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 21, Springer, 1990 | Zbl

[2] Diamond, Fred; Shurman, Jerry A first course in modular forms, Graduate Texts in Mathematics, 228, Springer, 2005 | MR | Zbl

[3] Drinfeld, V. G. Two theorems on modular curves, Funkts. Anal. Prilozh., Volume 7 (1973) no. 2, pp. 83-84 | MR | Zbl

[4] Etropolski, Anastassia; Morrow, Jackson; Zureick-Brown, David Sporadic torsion, 2016 (http://www.mathcs.emory.edu/~dzb/slides/DZB-SERMON-cubic-torsion.pdf)

[5] Frey, Gerhard Curves with infinitely many points of fixed degree, Isr. J. Math., Volume 85 (1994) no. 1, pp. 1-3 | MR | Zbl

[6] Igusa, Jun-Ichi Kroneckerian model of fields of elliptic modular functions, Am. J. Math., Volume 81 (1959), pp. 561-577 | DOI | MR | Zbl

[7] Ishii, N.; Momose, Fumiyuki Hyperelliptic modular curves, Tsukuba J. Math., Volume 15 (1991) no. 2, pp. 413-423 | DOI | MR | Zbl

[8] Jeon, Daeyeol; Kim, Chang Heon; Schweizer, Andreas On the torsion of elliptic curves over cubic number fields, Acta Arith., Volume 113 (2004) no. 3, pp. 291-301 | DOI | MR | Zbl

[9] Kamienny, Sheldon Torsion points on elliptic curves and q-coefficients of modular forms, Invent. Math., Volume 109 (1992) no. 2, pp. 221-229 | DOI | MR | Zbl

[10] Kato, Kazuya p-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-adiques et applications arithmétiques (Astérisque), Volume 295, Société Mathématique de France, 2004, pp. 117-290 | Numdam | Zbl

[11] Katz, Nicholas M. Galois properties of torsion points on abelian varieties, Invent. Math., Volume 62 (1981) no. 3, pp. 481-502 | DOI | MR | Zbl

[12] Kenku, M. A.; Momose, Fumiyuki Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J., Volume 109 (1988), pp. 125-149 | DOI | MR | Zbl

[13] Kubert, Daniel S. Universal bounds on the torsion of elliptic curves, Proc. Lond. Math. Soc., Volume 33 (1976) no. 2, pp. 193-237 | DOI | MR | Zbl

[14] Manin, Yu. I. Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 36 (1972), pp. 19-66 | MR | Zbl

[15] Mazur, Barry Modular curves and the Eisenstein ideal, Publ. Math., Inst. Hautes Étud. Sci., Volume 47 (1977), pp. 33-186 | DOI | Numdam | Zbl

[16] Merel, Loïc Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math., Volume 124 (1996) no. 1-3, pp. 437-449 | DOI | MR | Zbl

[17] Najman, Filip Torsion of rational elliptic curves over cubic fields and sporadic points on X 1 (n), Math. Res. Lett., Volume 23 (2016) no. 1, pp. 245-272 | DOI | MR | Zbl

[18] Ogg, Andrew P. Rational points on certain elliptic modular curves, Analytic Number Theory (Proceedings of Symposia in Pure Mathematics), Volume 1972, American Mathematical Society, 1972, pp. 221-231 | Zbl

[19] Ogg, Andrew P. Diophantine equations and modular forms, Bull. Am. Math. Soc., Volume 81 (1975), pp. 14-27 | MR | Zbl

[20] Parent, Pierre Torsion des courbes elliptiques sur les corps cubiques, Ann. Inst. Fourier, Volume 50 (2000) no. 3, pp. 723-749 | DOI | Numdam | MR | Zbl

[21] Parent, Pierre No 17-torsion on elliptic curves over cubic number fields, J. Théor. Nombres Bordeaux, Volume 15 (2003) no. 3, pp. 831-838 | DOI | MR | Zbl

[22] Raynaud, Michel Schémas en groupes de type (p,,p), Bull. Soc. Math. Fr., Volume 102 (1974), pp. 241-280 | DOI | MR | Zbl

[23] Shimura, Goro Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, 1, Mathematical Society of Japan, 1971 | MR | Zbl

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

[25] Tate, John; Oort, Frans Group schemes of prime order, Ann. Sci. Éc. Norm. Supér., Volume 3 (1970), pp. 1-21 | DOI | Numdam | MR | Zbl

[26] The Magma Development Team Magma (http://magma.maths.usyd.edu.au/magma/)

[27] Wang, Jian On the cyclic torsion of elliptic curves over cubic number fields, J. Number Theory, Volume 183 (2018), pp. 291-308 | DOI | MR | Zbl

[28] Waterhouse, William C. Abelian varieties over finite fields, Ann. Sci. Éc. Norm. Supér., Volume 2 (1969), pp. 521-560 | DOI | Numdam | MR | Zbl

Cité par Sources :