Generalizations of Iwasawa's 'Riemann-Hurwitz' Formula for Cyclic p-Extensions of Number Fields

TL;DR

This paper generalizes Iwasawa's Riemann-Hurwitz formula to cyclic p-extensions of number fields, deriving new congruences, inequalities, and a vanishing criterion for Iwasawa lambda-invariants.

Contribution

It extends Iwasawa's formula to higher degree cyclic p-extensions and introduces a vanishing criterion for lambda-invariants in broader contexts.

Findings

Derived new congruences and inequalities for cyclic p-extensions

Established a vanishing criterion for Iwasawa lambda-invariants

Generalized previous results for totally real fields

Abstract

We produce generalizations of Iwasawa's `Riemann-Hurwitz' formula for number fields. These generalizations apply to cyclic extensions of number fields of degree p^n for any positive integer n. We first deduce some congruences and inequalities and then use these formulas to establish a vanishing criterion for Iwasawa \lambda-invariants which generalizes a result of Takashi Fukuda et. al. for totally real number fields.