On characteristic elements modulo $p$ in non-commutative Iwasawa theory

TL;DR

This work develops a modulo $p$ theory for characteristic elements attached to Selmer groups in non-commutative Iwasawa theory by passing to localized $K_1$-groups and reducing modulo $p$ via a canonical map $\\pi_1$. It constructs the necessary Ore-localized algebras $\\mathcal{O}\\llbracket G \\rrbracket_{\\Sigma}$ and $k\\llbracket G \\rrbracket_{\\overline{\\Sigma}}$, and defines a modulo $p$ invariant that interpolates the integral characteristic element in a precise $K$-theoretic sense. The paper then applies this framework to Greenberg Selmer groups in non-commutative towers and to tensor products of modular forms with an Eisenstein prime, proving a non-commutative analogue of cyclotomic results: the mod-$p$ invariants multiply in the Eisenstein-twisted tensor product setting, and the mod-$p$ characteristic of certain Selmer groups factors according to the components arising from the Eisenstein prime. These results lay algebraic groundwork for a deeper connection with analytic aspects of the main conjecture and pave the way for future work on the modulo $p$-behavior of non-commutative main conjectures.

Abstract

Coates, Fukaya, Kato, Sujatha and Venjakob come up with a procedure of attaching suitable characteristic element to Selmer groups defined over a non-commutative $p$-adic Lie extension, which is subsequently refined by Burns and Venjakob. By their construction, these characteristic elements are realized as elements in an appropriate localized $K_1$-group. In this paper, we will introduce a notion of modulo $p$ for these elements and study some of their properties. As an application, we study the Greenberg Selmer group of a tensor product of modular forms, where $p$ is an Eisenstein prime for one of these forms.

Theorems & Definitions (40)

• Theorem 1.1
• Lemma 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Proposition 2.4
• proof
• Remark 2.5
• Lemma 2.6
• proof