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

For a finite abelian p-group A drank d.=dim A/pA, let M-A := log(p) vertical bar A vertical bar(1/d) be its (logarithmic) mean exponent. We study the behavior of the mean exponent of p-class groups in pro-p towers 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 M(G) attached to a finitely generated pro-p group G; when G = Gal(L/K), where L is the Hilbert p-class field tower of a number field K, M(G) measures the asymptotic behavior of the mean exponent of p-class groups inside 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.