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arXiv · 1603.01296

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

Fumio Sairaiji·Takuya Yamauchi
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Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ which has multiplicative reduction at a fixed prime $p$. For each positive integer $n$ we put $K_n:=\mathbb{Q}(E[p^n])$. The aim of this paper is to extend the author's previous our results 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 and describe the new lower bound in terms of the Mordell-Weil rank of $E(\mathbb{Q})$ and the ramification related to $E$.

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Fumio Sairaiji, Takuya Yamauchi. 2016-03-03. On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\mathbb{Q}$. https://arxiv.org/abs/1603.01296

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