On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\mathbb{Q}$

Sairaiji, Fumio; Yamauchi, Takuya

Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 893-915.

Let $E$ be an elliptic curve over $\mathbb{Q}$ 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 $K_n:=\mathbb{Q}\left(E\left[p^n\right]\right)$. 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 $E\left(\mathbb{Q}\right)$ and the ramification related to $E$.

Soit $E$ une courbe elliptique sur $\mathbb{Q}$ 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 $K_n:=\mathbb{Q}\left(E\left[p^n\right]\right)$. 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 $E\left(\mathbb{Q}\right)$ et de la ramification liée à $E$.

Classification: 11G05, 11G07
Keywords: elliptic curves, Mordell–Weil rank, class number

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