Triviality of Iwasawa module associated to some abelian fields of prime conductors

IF 0.4 4区 数学 Q4 MATHEMATICS
Humio Ichimura
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引用次数: 6

Abstract

Let p be an odd prime number and \(\ell \) an odd prime number dividing \(p-1\). We denote by \(F=F_{p,\ell }\) the real abelian field of conductor p and degree \(\ell \), and by \(h_F\) the class number of F. For a prime number \(r \ne p,\,\ell \), let \(F_{\infty }\) be the cyclotomic \(\mathbb {Z}_r\)-extension over F, and \(M_{\infty }/F_{\infty }\) the maximal pro-r abelian extension unramified outside r. We prove that \(M_{\infty }\) coincides with \(F_{\infty }\) and consequently \(h_F\) is not divisible by r when r is a primitive root modulo \(\ell \) and r is smaller than an explicit constant depending on p.

与一些素数导体阿贝尔场相关的Iwasawa模的平凡性
设p是奇质数 \(\ell \) 一个奇素数除法 \(p-1\)。我们用 \(F=F_{p,\ell }\) 导体p的实阿贝尔场和度 \(\ell \),和 \(h_F\) 对于素数f的类数 \(r \ne p,\,\ell \),让 \(F_{\infty }\) 做切眼手术 \(\mathbb {Z}_r\)-对F的扩展 \(M_{\infty }/F_{\infty }\) 我们证明了在r外无分支的最大亲r abel扩展 \(M_{\infty }\) 与…一致 \(F_{\infty }\) 因此 \(h_F\) 当r是原根模时不能被r整除 \(\ell \) r小于与p相关的显式常数。
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来源期刊
CiteScore
0.80
自引率
0.00%
发文量
7
审稿时长
>12 weeks
期刊介绍: The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.
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