Simultaneous indivisibility of class numbers of pairs of real quadratic fields

TL;DR

This paper proves that for certain integers t, a positive proportion of pairs of real quadratic fields have class numbers indivisible by 3, and demonstrates the existence of infinitely many such pairs with specific Iwasawa invariants, supporting Greenberg's conjecture.

Contribution

It establishes the existence of many pairs of real quadratic fields with class numbers indivisible by 3 for specific t, and links these results to Iwasawa theory and Greenberg's conjecture.

Findings

Positive proportion of quadratic field pairs with class numbers indivisible by 3 for t ≡ 0 mod 4

Infinitely many pairs with Iwasawa λ-invariant 0 for t ≡ 0 mod 12

Addresses a weak form of Iizuka's conjecture

Abstract

For a square-free integer $t$, Byeon \cite{byeon} 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 $t \geq 1$ 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 in \cite{iizuka}. As an application of our main result, we obtain that for any integer $t \geq 1$ 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 $K$.