Potential automorphy and the Leopoldt conjecture

TL;DR

This paper explores the non-abelian Leopoldt conjecture related to p-adic Hecke algebras over CM fields, using automorphy lifting techniques to connect it with classical Leopoldt conjectures for number fields.

Contribution

It proves an automorphy lifting theorem in the GL_n setting and shows how to deduce classical Leopoldt conjectures from the non-abelian conjecture without assuming automorphic induction.

Findings

Proved an automorphy lifting result conditional on Galois representations from torsion classes.

Connected the non-abelian Leopoldt conjecture to classical Leopoldt conjecture for totally real fields.

Demonstrated potential automorphy methods can derive classical conjectures without automorphic induction assumptions.

Abstract

We study in this paper Hida's p-adic Hecke algebra for GL_n over a CM field F. Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his conjecture in the case of F being the rationals implies the classical Leopoldt conjecture for a number field K of degree n over the rationals, if one assumes further the existence of automorphic induction of characters for the extension K over the rationals. We study Hida's conjecture using the automorphy lifting techniques adapted to the GL_n setting by Calegari--Geraghty. We prove an automorphy lifting result in this setting, conditional on existence and local-global compatibility of Galois representations arising from torsion classes in the cohomology of the corresponding symmetric manifolds. Under the same conditions we show that one can deduce the classical (abelian) Leopoldt conjectures for a totally real number field K and a prime p using Hida's non-abelian Leopoldt conjecture for p-adic Hecke algebra for GL_n over CM fields without needing to assume automorphic induction of characters for the extension K over the rationals. For this methods of potential automorphy results are used.