Rational torsion in elliptic curves and the cuspidal subgroup

Amod Agashe

doi:10.5802/jtnb.1017

Amod Agashe ¹

¹ Department of Mathematics Florida State University 1017 Academic Way Tallahassee, FL 32306, USA

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 81-91.

Résumé
Abstract

Soit $A$ une courbe elliptique sur $ℚ$ de conducteur $N$ sans facteurs carré, ayant un point rationnel d’ordre un nombre premier $r$ ne divisant pas $6 N$ . On montre alors que $r$ divise l’ordre du sous-groupe cuspidal $C$ de $J_{0} (N)$ . Si $A$ est une courbe de Weil, on peut la considérer comme une sous-variéte abélienne de $J_{0} (N)$ . Notre preuve montre plus precisément que $r$ divise l’ordre de $A \cap C$ . De plus, sous les hypothèses plus haut, mais sans supposer que $r$ ne divise pas $N$ , on montre qu’il existe un facteur premier $p$ de $N$ tel que la valeur propre de l’involution d’Atkin–Lehner $W_{p}$ agissant sur la forme modulaire associée à $A$ est égale à $- 1$ .

Let $A$ be an elliptic curve over $ℚ$ of square free conductor $N$ that has a rational torsion point of prime order $r$ such that $r$ does not divide $6 N$ . We show that then $r$ divides the order of the cuspidal subgroup $C$ of $J_{0} (N)$ . If $A$ is optimal, then viewing $A$ as an abelian subvariety of $J_{0} (N)$ , our proof shows more precisely that $r$ divides the order of $A \cap C$ . Also, under the hypotheses above minus the hypothesis that $r$ does not divide $N$ , we show that for some prime $p$ that divides $N$ , the eigenvalue of the Atkin–Lehner involution $W_{p}$ acting on the newform associated to $A$ is $- 1$ .

Reçu le : 2016-02-20
Accepté le : 2016-09-11
Accepté après révision le : 2017-01-29
Publié le : 2018-05-14

MR Zbl

DOI : 10.5802/jtnb.1017

Classification : 11G05, 14H52
Mots clés : Elliptic curves, torsion subgroup, cuspidal subgroup

Affiliations des auteurs :

Amod Agashe ¹

¹ Department of Mathematics Florida State University 1017 Academic Way Tallahassee, FL 32306, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2018__30_1_81_0,
     author = {Amod Agashe},
     title = {Rational torsion in elliptic curves and the cuspidal subgroup},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {81--91},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {1},
     year = {2018},
     doi = {10.5802/jtnb.1017},
     zbl = {1428.11104},
     mrnumber = {3809710},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1017/}
}

TY  - JOUR
AU  - Amod Agashe
TI  - Rational torsion in elliptic curves and the cuspidal subgroup
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 81
EP  - 91
VL  - 30
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1017/
DO  - 10.5802/jtnb.1017
LA  - en
ID  - JTNB_2018__30_1_81_0
ER  -

%0 Journal Article
%A Amod Agashe
%T Rational torsion in elliptic curves and the cuspidal subgroup
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 81-91
%V 30
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1017/
%R 10.5802/jtnb.1017
%G en
%F JTNB_2018__30_1_81_0

Amod Agashe. Rational torsion in elliptic curves and the cuspidal subgroup. Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 81-91. doi : 10.5802/jtnb.1017. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1017/

Bibliographie
Cité par

[1] Amod Agashe Conjectures concerning the orders of the torsion subgroup, the arithmetic component groups, and the cuspidal subgroup, Exp. Math., Volume 22 (2013) no. 4, pp. 363-366 | DOI | MR | Zbl

[2] Amod Agashe; William A. Stein Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Comput., Volume 74 (2005) no. 249, pp. 455-484 | DOI | MR | Zbl

[3] A. Oliver L. Atkin; Joseph Lehner Hecke operators on $Γ_{0} (m)$ , Math. Ann., Volume 185 (1970), pp. 134-160 | DOI

[4] Christophe Breuil; Brian Conrad; Fred Diamond; Richard Taylor On the modularity of elliptic curves over $ℚ$ : wild 3-adic exercises, J. Am. Math. Soc., Volume 14 (2001) no. 4, pp. 843-939 | DOI | MR | Zbl

[5] Seng-Kiat Chua; San Ling On the rational cuspidal subgroup and the rational torsion points of $J_{0} (p q)$ , Proc. Am. Math. Soc., Volume 125 (1997) no. 8, pp. 2255-2263 | DOI | MR | Zbl

[6] John E. Cremona Algorithms for modular elliptic curves, Cambridge University Press, 1997, 376 pages | Zbl

[7] Pierre Deligne; M. Rapoport Les schémas de modules de courbes elliptiques, Modular Functions of one Variable II, Proc. internat. Summer School, Univ. Antwerp 1972 (Lect. Notes Math.), Volume 349 (1973), pp. 143-316 | Zbl

[8] Fred Diamond; John Im Modular forms and modular curves, Seminar on Fermat’s last theorem (CMS Conf. Proc.), Volume 17 (1995), pp. 39-133 | MR | Zbl

[9] Neil Dummigan Rational torsion on optimal curves, Int. J. Number Theory, Volume 1 (2005) no. 4, pp. 513-531 | DOI | MR | Zbl

[10] Matthew Emerton Optimal quotients of modular Jacobians, Math. Ann., Volume 327 (2003) no. 3, pp. 429-458 | DOI | MR | Zbl

[11] Nicholas M. Katz $p$ -adic properties of modular schemes and modular forms, Modular Functions of one Variable III, Proc. internat. Summer School, Univ. Antwerp 1972 (Lect. Notes Math.), Volume 350 (1973), pp. 69-190 | MR | Zbl

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

[13] Masami Ohta Eisenstein ideals and the rational torsion subgroups of modular Jacobian varieties II, Tokyo J. Math., Volume 37 (2014) no. 2, pp. 273-318 | DOI | MR | Zbl

[14] William A. Stein The Cuspidal Subgroup of $J_{0} (N)$ (http://wstein.org/Tables/cuspgroup/index.html)

[15] Glenn Stevens Arithmetic on modular curves, Progress in Mathematics, 20, Birkhäuser, 1982, xvii+214 pages | MR | Zbl

[16] Glenn Stevens The cuspidal group and special values of $L$ -functions, Trans. Am. Math. Soc., Volume 291 (1985) no. 2, pp. 519-550 | MR | Zbl

[17] Shu-Leung Tang Congruences between modular forms, cyclic isogenies of modular elliptic curves and integrality of $p$ -adic $L$ -functions, Trans. Am. Math. Soc., Volume 349 (1997) no. 2, pp. 837-856 | DOI | MR | Zbl

[18] Vinayak Vatsal Multiplicative subgroups of $J_{0} (N)$ and applications to elliptic curves, J. Inst. Math. Jussieu, Volume 4 (2005) no. 2, pp. 281-316 | DOI | MR | Zbl

[19] Hwajong Yoo On Eisenstein ideals and the cuspidal group of $J_{0} (N)$ , Isr. J. Math., Volume 214 (2016) no. 1, pp. 359-377 | DOI | MR | Zbl

[20] Hwajong Yoo Rational torsion points on Jacobians of modular curves, Acta Arith., Volume 172 (2016) no. 4, pp. 299-304 | MR

Cité par Sources :