Simultaneous indivisibility of class numbers of pairs of real quadratic fields

Abstract

For a square-free integer t, Byeon (Proc. Am. Math. Soc. 132:3137–3140, 2004) proved the existence of infinitely many pairs of quadratic fields $\mathbb {Q}(\sqrt{D})$ and $\mathbb {Q}(\sqrt{tD})$ with $D > 0$ such that the class numbers of all of them are indivisible by 3. In the same spirit, we prove that for a given integer $t \ge 1$ with $t \equiv 0 \pmod {4}$, a positive proportion of fundamental discriminants $D > 0$ exist for which the class numbers of both the real quadratic fields $\mathbb {Q}(\sqrt{D})$ and $\mathbb {Q}(\sqrt{D + t})$ are indivisible by 3. This also addresses the complement of a weak form of a conjecture of Iizuka (J. Numb. Theory, 184:122–127, 2018). As an application of our main result, we obtain that for any integer $t \ge 1$ with $t \equiv 0 \pmod {12}$, there are infinitely many pairs of real quadratic fields $\mathbb {Q}(\sqrt{D})$ and $\mathbb {Q}(\sqrt{D + t})$ such that the Iwasawa $\lambda$-invariants associated with the basic $\mathbb {Z}_{3}$-extensions of both $\mathbb {Q}(\sqrt{D})$ and $\mathbb {Q}(\sqrt{D + t})$ are 0. For $p = 3$, this supports Greenberg’s conjecture which asserts that $\lambda _{p}(K) = 0$ for any prime number p and any totally real number field K.