Massey products and the Iwasawa theory of fine Selmer groups

TL;DR

This paper explores the relationship between Massey products, Galois cohomology, and Iwasawa theory, providing new characterizations of the vanishing of the $ ext{-}$-invariant for fine Selmer groups in the context of $p$-adic Galois representations.

Contribution

It introduces a novel cohomological framework using Massey products to analyze the $ ext{-}$-invariant and connects deformation ring properties with Iwasawa invariants.

Findings

Vanishing of the $ ext{-}$-invariant propagates to Greenberg neighborhoods.

Unobstructed deformation rings are equivalent to $ ext{-}$-invariant vanishing.

Connection established between Noetherian property of deformation rings and $ ext{-}$-invariant.

Abstract

A central conjecture of Coates and Sujatha predicts that the fine Selmer group of any $p$-adic Galois representation is cotorsion over the relevant Iwasawa algebra with vanishing $\mu$-invariant, generalizing Iwasawa's original conjecture for class groups. In this article, we recast this conjecture in terms of higher Galois cohomological operations called Massey products, stated purely in terms of the residual representation. This characterization implies that if the $\mu$-invariant vanishes for a given $\mathbb{Z}_p$-extension, then it also vanishes for all $\mathbb{Z}_p$-extensions the Greenberg neighbourhood of radius $1/p$. Furthermore, we establish that the unobstructedness of ordinary deformation rings over $\mathbb{Z}_p$-extensions is equivalent to the vanishing of the $\mu$-invariant of the fine Selmer group attached to the adjoint representation. For ordinary deformation rings attached to $2$-dimensional automorphic Galois representations, we establish a close connection between the Noetherian property and the vanishing of the $\mu$-invariant.