Genus of surcircular fields

TL;DR

This paper demonstrates that various Riemann-Hurwitz-style formulas for Iwasawa invariants in cyclotomic extensions are fundamentally equivalent, deriving from a common algebraic Herbrand quotient calculation in the cyclic prime degree case.

Contribution

It unifies multiple existing formulas for Iwasawa invariants by showing their algebraic equivalence through Herbrand quotient computations.

Findings

All formulas are algebraically equivalent.

The Herbrand quotient calculation suffices for the cyclic prime degree case.

Formulas hold for representations and purely algebraic contexts.

Abstract

We show that Riemann-Hurwitz-style translation formulas obtained by Kuz'min, Kida, Iwasawa, Wingberg et alii for the lambda invariant attached to certain Iwasawa moduli in cyclotomic Z{\ell}-extension of number fields are essentially equivalent. More precisely, we prove that all these formulas,including those stated in terms of representations, resul tidentically for purely algebraic reasons from the arithmetic computation of a suitable Herbrand quotient which it suffices to carry out in the cyclic case of prime degree {\ell}.