Class number divisibility of $\mathbb{Q}(\sqrt{3p}, \sqrt{m-21pn^{2}})$ constructed from elliptic curves of $2$-Selmer rank exactly $1$

TL;DR

This paper explores the connection between elliptic curves with specific Selmer ranks and the divisibility properties of class numbers in certain bi-quadratic number fields, providing explicit constructions and infinite families.

Contribution

It explicitly constructs unramified abelian extensions of bi-quadratic fields from elliptic curve points with 2-Selmer rank exactly 1, and demonstrates infinite families of fields with even class number.

Findings

Constructed unramified abelian extensions from elliptic curve points.

Established infinite families of bi-quadratic fields with even class number.

Linked elliptic curve properties to class number divisibility in number fields.

Abstract

The class number divisibility problem for number fields is one of the classical problems in algebraic number theory, which originated from Gauss' class number conjectures. The relation between the points on an elliptic curve and class number divisibility of a number field has been explored through the works of various mathematicians. Here, we explicitly construct an unramified abelian extension of a bi-quadratic field generated from points of a certain type of elliptic curve. Moreover, showing the $2$-Selmer rank of the said elliptic curve as $1$, we also construct an infinite family of bi-quadratic fields of even class number.