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

TL;DR

This paper explores the elliptic analogue of irregular primes by examining the divisibility properties of class numbers of division fields of specific elliptic curves, revealing infinite families with unique properties.

Contribution

It introduces a new class of elliptic curves with rank 1 whose division fields have class numbers divisible by high powers of primes, expanding understanding of elliptic analogues of irregular primes.

Findings

Constructed infinite families of elliptic curves with rank 1.

Proved class numbers of their division fields are divisible by p^{2n}.

Showed these division fields are pairwise non-isomorphic.

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 $p \geq 5$ 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.