No Riemann-hurwitz formula for the p-ranks of relative class groups

TL;DR

This paper demonstrates through numerical examples that a Riemann-Hurwitz type formula does not hold for p-ranks of relative class groups in certain p-extensions of CM-type number fields, and explores related Galois structures.

Contribution

It disproves the existence of a Riemann-Hurwitz formula for p-ranks in these extensions and analyzes the conditions under which Kida's lambda invariant formula applies.

Findings

Numerical examples show the non-existence of a Riemann-Hurwitz formula for p-ranks.

Certain theoretical group structures are proven not to exist in specific cases.

Kida's formula for lambda invariants holds only when the p-class group reduces to ambiguous classes.

Abstract

We disprove, by means of numerical examples, the existence of a Riemann-Hurwitz formula for the p-ranks of relative class groups in a p-ramified p-extension K/k of number fields of CM-type containing ?\_p. In the cyclic case of degree p, under some assumptions on the p-class group of k, we prove some properties of the Galois structure of the p-class group of K; but we have found, through numerical experimentation, that some theoretical group structures do not exist in this particular situation, and we justify this fact. Then we show, in this context, that Kida's formula on lambda invariants is valid for the p-ranks if and only if the p-class group of K is reduced to the group of ambiguous classes, which is of course not always the case.