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On the Invariant Factors of Class Groups in Towers of Number Fields

Published online by Cambridge University Press:  20 November 2018

Farshid Hajir  and
Christian Maire
Farshid Hajir
Affiliation:
Department of Mathematics & Statistics, University of Massachusetts, Amherst MA 01003, USA e-mail: [email protected]
Christian Maire
Affiliation:
Laboratoire de Mathématiques, Université Bourgogne Franche-Comté et CNRS, (UMR 6623), 16 route deGray, 25030 Besançon cédex, France e-mail: [email protected]
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Abstract

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For a finite abelian $p$-group $A$ of rank $d\,=\,\dim\,A/pA$, let ${{\mathbb{M}}_{A}}\,:=\,\text{lo}{{\text{g}}_{p}}\,{{\left| A \right|}^{1/d}}$ be its (logarithmic) mean exponent. We study the behavior of the mean exponent of $p$-class groups in pro-$p$ towers $\text{L/K}$ 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 $\underline{\mathbb{M}}\left( G \right)$ attached to a finitely generated pro-$p$ group $G$; when $G\,=\,\text{Gal}\left( \text{L/K} \right)$, where $L$ is the Hilbert $p$-class field tower of a number field $K$, $\underline{\mathbb{M}}\left( G \right)$ measures the asymptotic behavior of the mean exponent of $p$-class groups inside $\text{L/K}$. 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

11R2911R37class field towerideal class grouppro-p groupp-adic analytic groupBrauer-Siegel Theorem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 1 , 01 February 2018 , pp. 142 - 172
Copyright
Copyright © Canadian Mathematical Society 2018

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