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Graduated orders over completed group rings and conductor formulæ

Ben Forrás

TL;DR

This work extends the theory of conductors and Artin-type conjectures from classical maximal/hereditary orders to graduated orders over completed group rings of one-dimensional $p$-adic Lie groups. By developing a two-dimensional base framework, it proves a conductor formula for graduated overorders, identifies an explicit central-conductor expression with componentwise $r_\chi$ exponents determined by ramification data, and refines Nickel’s results in this broader setting. The paper also confirms the equivariant $p$-adic Artin conjecture for graduated orders assuming the equivariant Iwasawa main conjecture, with a componentwise reduction via a detailed Wedderburn decomposition. Collectively, these results broaden the toolkit for studying non-Dedekind base rings in noncommutative Iwasawa theory, providing explicit invariants, duality properties, and annihilation results for Ext- and Fitting-type invariants in graduated contexts.

Abstract

We study graduated orders over completed group rings of $1$-dimensional admissible $p$-adic Lie groups, and verify the equivariant $p$-adic Artin conjecture for such orders. Following Jacobinski and Plesken, we obtain a formula for the conductor of a graduated order into a self-dual order. We also refine Nickel's central conductor formula by determining a hitherto implicit exponent $r_χ$.

Graduated orders over completed group rings and conductor formulæ

TL;DR

This work extends the theory of conductors and Artin-type conjectures from classical maximal/hereditary orders to graduated orders over completed group rings of one-dimensional -adic Lie groups. By developing a two-dimensional base framework, it proves a conductor formula for graduated overorders, identifies an explicit central-conductor expression with componentwise exponents determined by ramification data, and refines Nickel’s results in this broader setting. The paper also confirms the equivariant -adic Artin conjecture for graduated orders assuming the equivariant Iwasawa main conjecture, with a componentwise reduction via a detailed Wedderburn decomposition. Collectively, these results broaden the toolkit for studying non-Dedekind base rings in noncommutative Iwasawa theory, providing explicit invariants, duality properties, and annihilation results for Ext- and Fitting-type invariants in graduated contexts.

Abstract

We study graduated orders over completed group rings of -dimensional admissible -adic Lie groups, and verify the equivariant -adic Artin conjecture for such orders. Following Jacobinski and Plesken, we obtain a formula for the conductor of a graduated order into a self-dual order. We also refine Nickel's central conductor formula by determining a hitherto implicit exponent .

Paper Structure

This paper contains 21 sections, 30 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Overview of conductor formulæ
  3. New results
  4. Notation and terminology
  5. Acknowledgements
  6. Graduated orders in completed group rings
  7. Inadequacy of hereditary orders over Iwasawa algebras
  8. Graduated orders over two-dimensional rings
  9. Standard form
  10. $\mathbbold \Lambda$-lattices
  11. Inverse differents
  12. Intersections of maximal orders
  13. The $(\cap,+)$-lattice of $\mathbbold \Lambda$-lattices
  14. Extremal orders
  15. Graduated hulls
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $H$ be a finite group, and let $R$ be a Dedekind domain with field of fractions $L=\mathop{\mathrm{Frac}}\nolimits(R)$ such that the characteristic of $L$ is coprime to $\#H$. Let $\mathbbold \Gamma$ be a maximal $R$-order in the group algebra $L[H]$ containing $R[H]$. Then

Theorems & Definitions (75)

  • Theorem 1.1: Jacobinski, CR
  • Theorem 1.2: NickelConductor
  • Theorem 1.3: Jacobinski, CR
  • Theorem 1.4: \ref{['general-conductor-formula']}
  • Theorem 1.5: §§\ref{['sec:conductor-formula']}--\ref{['sec:central-conductor']}
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 65 more