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Kida's formula via Selmer complexes

Takenori Kataoka

Abstract

In Iwasawa theory, the $λ$, $μ$-invariants of various arithmetic modules are fundamental invariants that measure the size of the modules. Concerning the minus components of the unramified Iwasawa modules, Kida proved a formula that describes the behavior of those invariants with respect to field extensions. Subsequently, many analogues of Kida's formula have been found in various areas in Iwasawa theory. In this paper, we present a novel approach to such analogues of Kida's formula, based on the perspective of Selmer complexes.

Kida's formula via Selmer complexes

Abstract

In Iwasawa theory, the , -invariants of various arithmetic modules are fundamental invariants that measure the size of the modules. Concerning the minus components of the unramified Iwasawa modules, Kida proved a formula that describes the behavior of those invariants with respect to field extensions. Subsequently, many analogues of Kida's formula have been found in various areas in Iwasawa theory. In this paper, we present a novel approach to such analogues of Kida's formula, based on the perspective of Selmer complexes.
Paper Structure (23 sections, 20 theorems, 60 equations)

This paper contains 23 sections, 20 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The algebraic theorem
  4. Kida's formula for ordinary representations
  5. Arithmetic applications
  6. Analogues for $p$-adic Lie extensions
  7. Review of previous work
  8. The key algebraic theorem
  9. Module version
  10. Complex version
  11. Application to ordinary representations
  12. Statement of the theorem
  13. The Selmer complex
  14. Proof of Theorem \ref{['thm:main_ord']}
  15. Arithmetic applications
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $G$ be a finite $p$-group. Let $C$ be a perfect complex over the group ring $\Lambda[G]$, and put $\overline{C} = \Lambda \otimes^{\mathsf{L}}_{\Lambda[G]} C$. Then $\mu = 0$ for $C$ is equivalent to $\mu = 0$ for $\overline{C}$. If these equivalent conditions hold, then we have $\lambda(C) = (\

Theorems & Definitions (42)

  • Theorem 1.1: Theorem \ref{['thm:Kida_alg']}
  • Theorem 1.2: Theorem \ref{['thm:main_ord']}
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • ...and 32 more