Rational torsion in elliptic curves and the cuspidal subgroup

Amod Agashe

doi:10.5802/jtnb.1017

Amod Agashe ¹

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

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

Abstract (Original language)
Résumé

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 $- 1$ .

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 $- 1$ .

Article information
Cite this article

Received: 2016-02-20
Accepted: 2016-09-11
Revised after acceptance: 2017-01-29
Published online: 2018-05-14

MR Zbl

DOI: 10.5802/jtnb.1017

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

Author's affiliations:

Amod Agashe ¹

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

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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

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