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# On the invariant factors of class groups in towers of number fields

Published:2017-11-02
Printed: Feb 2018
• Farshid Hajir,
Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003, USA
• Christian Maire,
Laboratoire de Mathématiques, Université Bourgogne Franche-Comté et CNRS (UMR 6623), 16 route de Gray, 25030 Besançon cédex, France
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## Abstract

For a finite abelian $p$-group $A$ of rank $d=\dim A/pA$, let $\mathbb{M}_A := \log_p |A|^{1/d}$ be its \emph{(logarithmic) mean exponent}. We study the behavior of the mean exponent of $p$-class groups in pro-$p$ towers $\mathrm{L}/K$ of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-$p$ towers in which the mean exponent of $p$-class groups remains bounded. Several explicit examples are given with $p=2$. Turning to group theory, we introduce an invariant $\underline{\mathbb{M}}(G)$ attached to a finitely generated pro-$p$ group $G$; when $G=\operatorname{Gal}(\mathrm{L}/\mathrm{K})$, where $\mathrm{L}$ is the Hilbert $p$-class field tower of a number field $K$, $\underline{\mathbb{M}}(G)$ measures the asymptotic behavior of the mean exponent of $p$-class groups inside $\mathrm{L}/\mathrm{K}$. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.
 Keywords: class field tower, ideal class group, pro-p group, p-adic analytic group, Brauer-Siegel Theorem
 MSC Classifications: 11R29 - Class numbers, class groups, discriminants 11R37 - Class field theory

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