论文标题
对真实二次字段的班级数量的同时不可分割
Simultaneous indivisibility of class numbers of pairs of real quadratic fields
论文作者
论文摘要
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 $。本着同样的精神，我们证明，对于给定的整数$ t \ geq 1 $，带有$ t \ equiv 0 \ pmod {4} $，基本歧视剂$ d> 0 $的正比例为$ d> 0 $，这两个真正的Quadratic Fields $ \ Mathbb {Q}（Q}（Q}（Q}（\ sqrt））的类数$ \ mathbb {q}（\ sqrt {d + t}）$不可分解$ 3 $。这也解决了\ cite {iizuka}中iizuka猜想的弱形式的补充。作为我们主要结果的应用，我们可以获得任何整数$ t \ geq 1 $带有$ t \ equiv 0 \ equiv 0 \ pmod {12} $，有无限的真实二次次数$ \ mathbb {q}（q}（q}（q}（\ sqrt {d}） Iwasawa $λ$ -Invariants与基本$ \ MATHBB {Z} _ {3} $ - 两种$ \ Mathbb {q}（\ sqrt {d}）$和$ \ \ \ \ \ \ \ m athbb {q}（Q}（\ sqrt {\ sqrt {d + t + t}）$ 0 $ 0的扩展。对于$ p = 3 $，这支持了格林伯格的猜想，该猜想断言$λ_{p}（k）（k）= 0 $对于任何质数$ p $和任何完全实数的字段$ k $。
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 $λ$-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 $λ_{p}(K) = 0$ for any prime number $p$ and any totally real number field $K$.
