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

TL;DR

This paper investigates the behavior of the mean exponent of p-class groups in towers of number fields, introducing a new invariant to measure their asymptotic growth and constructing examples where this mean remains bounded.

Contribution

It introduces the invariant M(G) for finitely generated FAb pro-p groups and applies it to analyze class group behavior in number field towers, combining group theory with number theory techniques.

Findings

Constructed infinite tamely ramified towers with bounded mean exponent

Introduced the invariant M(G) to measure asymptotic growth of class groups

Provided explicit examples with p=2

Abstract

For a finite abelian p-group A of rank d, we define its (logarithmic) mean exponent to be the base-p logarithm of the d-th root of its cardinality. We study the behavior of the mean exponent of p-class groups in towers of number fields. By combining techniques from group theory with the Tsfasman-Valdut generalization of the Brauer-Siegel Theorem, we construct infinite tamely ramified towers in which the mean exponent of class groups remains bounded. Several explicit examples with p=2 are given. We introduce an invariant M(G) attached to a finitely generated FAb pro-p group G which measures the asymptotic growth of the mean exponent of abelianizations of subgroups of index n with n going to infinity. When G=Gal(L/K), M(G) measures the asymptotic behavior of the mean exponent of class groups in 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.