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On the Iwasawa invariants of Artin representations

Aditya Karnataki, Anwesh Ray

TL;DR

The paper investigates Iwasawa invariants $\mu_p$ and $\lambda_p$ attached to Selmer groups $S_{\chi,\epsilon}(\mathbb{Q}_\infty)$ of irreducible Artin representations. It develops an Euler-characteristic framework and control theorems (Greenberg–Vatsal) to obtain explicit vanishing criteria for these invariants, connecting them to $p$-rationality of the base field and to distribution questions for primes. Special cases in low-dimensional Artin representations (dihedral 2D and icosahedral 3D) illustrate concrete vanishing criteria, while distribution results tie the arithmetic of $p$-rational primes to the behavior of Selmer groups. The work further links classical Iwasawa theory of class groups with Selmer-theoretic phenomena for Artin representations, offering heuristic densities and unconditional results in certain $S_3$-extensions through conjectures like Gras and the Maire–Rougnant findings on $p$-rationality.

Abstract

We study Iwasawa invariants associated to Selmer groups of Artin representations, and criteria for the vanishing of the associated algebraic Iwasawa invariants. The conditions obtained can be used to study natural distribution questions in this context.

On the Iwasawa invariants of Artin representations

TL;DR

The paper investigates Iwasawa invariants and attached to Selmer groups of irreducible Artin representations. It develops an Euler-characteristic framework and control theorems (Greenberg–Vatsal) to obtain explicit vanishing criteria for these invariants, connecting them to -rationality of the base field and to distribution questions for primes. Special cases in low-dimensional Artin representations (dihedral 2D and icosahedral 3D) illustrate concrete vanishing criteria, while distribution results tie the arithmetic of -rational primes to the behavior of Selmer groups. The work further links classical Iwasawa theory of class groups with Selmer-theoretic phenomena for Artin representations, offering heuristic densities and unconditional results in certain -extensions through conjectures like Gras and the Maire–Rougnant findings on -rationality.

Abstract

We study Iwasawa invariants associated to Selmer groups of Artin representations, and criteria for the vanishing of the associated algebraic Iwasawa invariants. The conditions obtained can be used to study natural distribution questions in this context.
Paper Structure (14 sections, 26 theorems, 76 equations)

This paper contains 14 sections, 26 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Iwasawa theory of class groups and Artin representations
  4. Classical Iwasawa theory and Iwasawa invariants
  5. The leading coefficient of the characteristic series
  6. Artin representations
  7. The Euler characteristic
  8. Definition and properties of the Euler characteristic
  9. Calculating the Euler characteristic
  10. The structure of the Selmer group $S_{\chi, \epsilon}(\mathbb{Q})$
  11. Special cases
  12. Distribution questions for Artin representations
  13. Iwasawa theory of class groups
  14. Artin representations

Key Result

Theorem 1.2

Suppose that the Assumption main ass holds. Then, the Selmer group $S_{\chi, \epsilon}(\mathbb{Q}_n)$ is finite for all $n\in \mathbb{Z}_{\geq 0}$.

Theorems & Definitions (59)

  • Theorem 1.2: Greenberg and Vatsal
  • proof
  • Definition 1.3
  • Theorem 1.4: Theorem \ref{['cor 3.10']}
  • Theorem 1.6: Theorem \ref{['last theorem']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • ...and 49 more