On the simultaneous $3$-divisibility of class numbers of quadruples of real quadratic fields

TL;DR

This paper constructs infinitely many quadruples of real quadratic fields with class numbers divisible by 3, marking a novel advance in understanding class number divisibility and its applications to elliptic curves.

Contribution

It provides the first known construction of infinitely many quadruples of real quadratic fields with all class numbers divisible by 3.

Findings

Constructed infinitely many quadruples with class numbers divisible by 3

First result on divisibility of class numbers in such tuples

Applied findings to produce elliptic curves with 3-torsion subgroup

Abstract

In this paper, we construct infinitely many quadruples of real quadratic fields whose class numbers are all divisible by $3$. To the best of our knowledge, this is the first result towards the divisibility of the class numbers of certain tuples of real quadratic fields. At the end, we give an application of this result to produce some elliptic curves having a $3$-torsion subgroup.