On the class numbers of the fields of the p n -torsion points of elliptic curves over
Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 893-915.

Soit E une courbe elliptique sur ayant réduction multiplicative en un nombre premier p. Supposons que en tout nombre premier différent de p la courbe E a une réduction multiplicative ou potentiellement bonne. Pour chaque entier positif n on pose K n :=(E[p n ]). Le but de cet article est d’étendre nos résultats précédents [13] concernant l’ordre du p-sous-groupe de Sylow du groupe des classes d’idéaux de K n à un cadre plus général. Nous modifions également la borne inférieure précédente de cet ordre donnée en termes du rang de Mordell–Weil de E() et de la ramification liée à E.

Let E be an elliptic curve over which has multiplicative reduction at a fixed prime p. Assume E has multiplicative reduction or potentially good reduction at any prime not equal to p. For each positive integer n we put K n :=(E[p n ]). The aim of this paper is to extend the authors’ previous results in [13] concerning with the order of the p-Sylow group of the ideal class group of K n to more general setting. We also modify the previous lower bound of the order given in terms of the Mordell–Weil rank of E() and the ramification related to E.

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DOI : 10.5802/jtnb.1056
Classification : 11G05, 11G07
Mots clés : elliptic curves, Mordell–Weil rank, class number
Fumio Sairaiji 1 ; Takuya Yamauchi 2

1 Faculty of Nursing, Hiroshima International University, Hiro, Hiroshima 737-0112, Japan
2 Mathematical Institute, Tohoku University 6-3, Aoba, Aramaki, Aoba-Ku, Sendai 980-8578, Japan
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Fumio Sairaiji; Takuya Yamauchi. On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\protect \mathbb{Q}$. Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 893-915. doi : 10.5802/jtnb.1056. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1056/

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