Journal of the Mathematical Society of Japan

The $l$-class group of the Z$_p$-extension over the rational field

Kuniaki HORIE and Mitsuko HORIE

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Let $p$ be an odd prime, and let B$_\infty$ denote the Z$_p$-extension over the rational field. Let $l$ be an odd prime different from $p$. The question whether the $l$-class group of B$_\infty$ is trivial has been considered in our previous papers mainly for the case where $l$ varies with $p$ fixed. We give a criterion, for checking the triviality of the $l$-class group of B$_\infty$, which enables us to discuss the triviality when $p$ varies with $l$ fixed. As a consequence, we find that, if $l$ does not exceed 13 and $p$ does not exceed 101, then the $l$-class group of B$_\infty$ is trivial.

Article information

J. Math. Soc. Japan, Volume 64, Number 4 (2012), 1071-1089.

First available in Project Euclid: 29 October 2012

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Zentralblatt MATH identifier

Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R20: Other abelian and metabelian extensions 11R23: Iwasawa theory

Zp-extension l-class group reflection theorem class number formula Iwasawa theory


HORIE, Kuniaki; HORIE, Mitsuko. The $l$-class group of the Z $_p$-extension over the rational field. J. Math. Soc. Japan 64 (2012), no. 4, 1071--1089. doi:10.2969/jmsj/06441071.

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