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, Volume 30 (2018) no. 1, pp. 81-91.

Abstract
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 $- 1$ .

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
Mots-clés : 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

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     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},
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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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1017/

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