发表机构
Rikkyo University（立教大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出非分歧Iwasawa模有限性判据，构造无穷族实二次域，其类群$2$-秩为三且分圆$\mathbb{Z}_2$-扩张完全分歧并满足Greenberg猜想。
AI 中文摘要
本文研究实二次域$\mathbb{Q}(\sqrt{p_1p_2p_3p_4})$上的分圆$\mathbb{Z}_2$-扩张的非分歧Iwasawa模，其中$p_1, p_2, p_3, p_4$为互不相同的奇素数。我们给出了数域的非分歧Iwasawa模有限性的一个判据，该判据基于具有特定商群的$2$-群的不存在性。利用此判据，我们构造了一个无穷族这样的实二次域，其非分歧Iwasawa模的类型为$\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$，$2$-类群类型为$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$。这给出了第一个无穷族实二次域的例子，其理想类群的$2$-秩为三，且其分圆$\mathbb{Z}_2$-扩张是完全分歧的并满足Greenberg猜想。
英文摘要
In this paper, we study the unramified Iwasawa module over the cyclotomic $\mathbb{Z}_2$-extension of the real quadratic field $\mathbb{Q}(\sqrt{p_1p_2p_3p_4})$, where $p_1, p_2, p_3$, and $p_4$ are distinct odd prime numbers. We give a criterion for the finiteness of an unramified Iwasawa module of a number field. The criterion is based on the nonexistence of a $2$-group with a certain prescribed quotient. Using this criterion, we construct an infinite family of such real quadratic fields with unramified Iwasawa module of type $\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ and $2$-class group of type $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. This gives the first example of an infinite family of real quadratic fields whose ideal class group has $2$-rank three and whose cyclotomic $\mathbb{Z}_2$-extension is totally ramified and satisfies Greenberg's conjecture.
Comments17 pages, 1 figure, 4 tables
