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 : https://doi.org/10.5802/jtnb.1056
Classification : 11G05,  11G07
Mots clés : elliptic curves, Mordell–Weil rank, class number
@article{JTNB_2018__30_3_893_0,
     author = {Fumio Sairaiji and Takuya Yamauchi},
     title = {On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\protect \mathbb{Q}$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {893--915},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1056},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1056/}
}
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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