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

Hajir, Farshid ; Maire, Christian

Canadian journal of mathematics, Tome 70 (2018) no. 1, pp. 142-172

Résumé

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.

Détail

DOI : 10.4153/CJM-2017-032-9

Mots-clés : 11R29, 11R37, class field tower, ideal class group, pro-p group, p-adic analytic group, Brauer-Siegel Theorem