On the cyclic torsion of elliptic curves over cubic number fields (II)

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 $\mathbb{Z}/N\mathbb{Z}$ n’est pas un sous-groupe de $E{\left(K\right)}_{\mathrm{tor}}$ 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 $\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E{\left(K\right)}_{\mathrm{tor}}$ for any elliptic curve $E$ over a cubic number field $K$.

[1] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 21, Springer, 1990 | Zbl 0705.14001

[2] Diamond, Fred; Shurman, Jerry A first course in modular forms, Graduate Texts in Mathematics, 228, Springer, 2005 | MR 2112196 | Zbl 1062.11022

[3] Drinfeld, V. G. Two theorems on modular curves, Funkts. Anal. Prilozh., Volume 7 (1973) no. 2, pp. 83-84 | MR 318157 | Zbl 0285.14006

[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 1264340 | Zbl 0808.14022

[6] Igusa, Jun-Ichi Kroneckerian model of fields of elliptic modular functions, Am. J. Math., Volume 81 (1959), pp. 561-577 | Article | MR 108498 | Zbl 0093.04502

[7] Ishii, N.; Momose, Fumiyuki Hyperelliptic modular curves, Tsukuba J. Math., Volume 15 (1991) no. 2, pp. 413-423 | Article | MR 1138196 | Zbl 0771.14008

[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 | Article | MR 2069117 | Zbl 1083.11038

[9] Kamienny, Sheldon Torsion points on elliptic curves and q-coefficients of modular forms, Invent. Math., Volume 109 (1992) no. 2, pp. 221-229 | Article | MR 1172689 | Zbl 0773.14016

[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 1142.11336

[11] Katz, Nicholas M. Galois properties of torsion points on abelian varieties, Invent. Math., Volume 62 (1981) no. 3, pp. 481-502 | Article | MR 604840 | Zbl 0471.14023

[12] Kenku, M. A.; Momose, Fumiyuki Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J., Volume 109 (1988), pp. 125-149 | Article | MR 931956 | Zbl 0647.14020

[13] Kubert, Daniel S. Universal bounds on the torsion of elliptic curves, Proc. Lond. Math. Soc., Volume 33 (1976) no. 2, pp. 193-237 | Article | MR 434947 | Zbl 0331.14010

[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 314846 | Zbl 0243.14008

[15] Mazur, Barry Modular curves and the Eisenstein ideal, Publ. Math., Inst. Hautes Étud. Sci., Volume 47 (1977), pp. 33-186 | Article | Numdam | Zbl 0394.14008

[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 | Article | MR 1369424 | Zbl 0936.11037

[17] Najman, Filip Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1\left(n\right)$, Math. Res. Lett., Volume 23 (2016) no. 1, pp. 245-272 | Article | MR 3512885 | Zbl 1416.11084

[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 0273.14008

[19] Ogg, Andrew P. Diophantine equations and modular forms, Bull. Am. Math. Soc., Volume 81 (1975), pp. 14-27 | MR 354675 | Zbl 0316.14012

[20] Parent, Pierre Torsion des courbes elliptiques sur les corps cubiques, Ann. Inst. Fourier, Volume 50 (2000) no. 3, pp. 723-749 | Article | Numdam | MR 1779891 | Zbl 0971.11030

[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 | Article | MR 2142238 | Zbl 1072.11037

[22] Raynaud, Michel Schémas en groupes de type $(p,\cdots ,p)$, Bull. Soc. Math. Fr., Volume 102 (1974), pp. 241-280 | Article | MR 419467 | Zbl 0325.14020

[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 314766 | Zbl 0221.10029

[24] Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, Springer, 1994 | MR 1312368 | Zbl 0911.14015

[25] Tate, John; Oort, Frans Group schemes of prime order, Ann. Sci. Éc. Norm. Supér., Volume 3 (1970), pp. 1-21 | Article | Numdam | MR 265368 | Zbl 0195.50801

[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 | Article | MR 3715238 | Zbl 06802535

[28] Waterhouse, William C. Abelian varieties over finite fields, Ann. Sci. Éc. Norm. Supér., Volume 2 (1969), pp. 521-560 | Article | Numdam | MR 265369 | Zbl 0188.53001