Elliptic analogue of irregular prime numbers for the $p^n$-division fields of the curves $y^2=x^3-(s^4+t^2)x$

Abstract

A prime number $p$ is said to be irregular if it divides the class number of the $p$-th cyclotomic field $\mathbb{Q}({\zeta }_p)=\mathbb{Q}({\mathbb{G}}_m[p])$. In this paper, we study its elliptic analogue for the division fields of an elliptic curve. More precisely, for a prime number $p\ge 5$ and a positive integer $n$, we study the $p$-divisibility of the class number of the $p^n$-division field $\mathbb{Q}(E[p^n])$ of an elliptic curve $E$ of the form $y^2=x^3-(s^4+t^2)x$. In particular, we construct a certain infinite subfamily consisting of curves with novel properties that they are of Mordell-Weil rank 1 and the class numbers of their $p^n$-division fields are divisible by $p^{2n}$. Moreover, we can prove that these division fields are not isomorphic to each other. In our construction, we use recent results obtained by the first author.

15 pages, comments welcome!