Noncommutative Iwasawa theory arising from Hecke algebras

TL;DR

This paper develops a noncommutative Iwasawa theory framework for Hilbert modular forms over totally real fields, introducing new deformation rings and formulating a main conjecture in this setting.

Contribution

It introduces a new category of modules for noncommutative Iwasawa theory and formulates a main conjecture for Selmer groups in this context.

Findings

Proves a control theorem for deformation rings over $p$-adic Lie extensions.

Defines a category of torsion modules generalizing Venjakob's Ore set.

Formulates a noncommutative main conjecture for Selmer groups.

Abstract

Let $p$ be an odd prime and $f$ be a nearly ordinary Hilbert modular Hecke eigenform defined over a totally real field $F$. Let $\mathbb{I}$ be an irreducible component of the universal nearly ordinary or locally cyclotomic deformation of the representation of $\mathrm{Gal}_F$ that is associated to $f$. We study the deformation rings over a $p$-adic Lie extension $F_\infty$ that contains the cyclotomic $\mathbb{Z}_p$-extension of $F$. More precisely, we prove a control theorem about these rings. We introduce a category $\mathfrak{M}_{\mathcal H}^{\mathbb I}(\mathcal G)$, where $\mathcal G=\mathrm{Gal}(F_\infty/F)$ and $\mathcal H=\mathrm{Gal}(F_\infty/F_{cyc})$, which is the category of modules which are torsion with respect to a certain Ore set, which generalizes the Ore set introduced by Venjakob. For Selmer groups which are in this category, we formulate a Main conjecture in the spirit of Noncommutative Iwasawa theory. We then set up a strategy to prove the conjecture by generalizing work of Burns, Kato, Kakde, and Ritter and Weiss. This requires appropriate generalizations of results of Oliver and Taylor, and Oliver on Logarithms of certain $K$-groups, which we have presented here.