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

TL;DR

This paper explores the properties of characteristic elements modulo p in non-commutative Iwasawa theory, focusing on their algebraic structure and applications to Selmer groups of modular forms, especially when p is an Eisenstein prime.

Contribution

It introduces a notion of modulo p for characteristic elements in non-commutative Iwasawa theory and studies their properties and applications.

Findings

Defined modulo p for characteristic elements.

Analyzed properties of these elements.

Applied to Greenberg Selmer groups of modular forms.

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.