Rational torsion in elliptic curves and the cuspidal subgroup
Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 81-91.

Soit A une courbe elliptique sur de conducteur N sans facteurs carré, ayant un point rationnel d’ordre un nombre premier r ne divisant pas 6N. 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 AC. 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  of square free conductor N that has a rational torsion point of prime order r such that r does not divide 6N. 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 AC. 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.

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DOI : 10.5802/jtnb.1017
Classification : 11G05, 14H52
Mots clés : Elliptic curves, torsion subgroup, cuspidal subgroup
Agashe, Amod 1

1 Department of Mathematics Florida State University 1017 Academic Way Tallahassee, FL 32306, USA
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Agashe, Amod. 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. http://archive.numdam.org/articles/10.5802/jtnb.1017/

[1] Agashe, Amod 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] Agashe, Amod; Stein, William A. 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] Atkin, A. Oliver L.; Lehner, Joseph Hecke operators on Γ 0 (m), Math. Ann., Volume 185 (1970), pp. 134-160 | DOI

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

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

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

[7] Deligne, Pierre; Rapoport, M. 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, Springer (1973), pp. 143-316 | Zbl

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

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

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

[11] Katz, Nicholas M. 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, Springer (1973), pp. 69-190 | MR | Zbl

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

[13] Ohta, Masami 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] Stein, William A. The Cuspidal Subgroup of J 0 (N) (http://wstein.org/Tables/cuspgroup/index.html)

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

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

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

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