Some remarks on Kida's formula when $\mu\neq 0$

TL;DR

This paper discusses the extension of Kida's formula in Iwasawa theory to cases where the $dmu$-invariant is non-zero, proposing a conjectural explanation based on noncommutative Iwasawa theory.

Contribution

It offers a conceptual, though conjectural, explanation for the validity of Kida's formula when $dmu eq 0$, expanding the theoretical framework.

Findings

Proposes a conjectural extension of Kida's formula to $dmu eq 0$ cases.

Links the extension to the $\u012bm_{H}(G)$-conjecture in noncommutative Iwasawa theory.

Provides a conceptual understanding rather than a proven result.

Abstract

The Kida's formula in classical Iwasawa theory relates the Iwasawa $\lambda$-invariants of $p$-extensions of number fields. Analogue of this formula was subsequently established for the Iwasawa $\lambda$-invariants of Selmer groups under an appropriate $\mu=0$ assumption. In this paper, we give a conceptual (but conjectural) explanation that such a formula should also hold when $\mu\neq 0$. The conjectural component comes from the so-called $\mathfrak{M}_H(G)$-conjecture in noncommutative Iwasawa theory.