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

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.

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.

Received:
Accepted:
Revised after acceptance:
Published online:
DOI: 10.5802/jtnb.1017
Classification: 11G05, 14H52
Keywords: 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, Volume 30 (2018) no. 1, pp. 81-91. doi : 10.5802/jtnb.1017. http://www.numdam.org/articles/10.5802/jtnb.1017/

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