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

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.