Potential Automorphy and the Leopoldt conjecture


We study in this paper Hida's $p$-adic Hecke algebra for ${\rm GL}_n$ 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 $F={\Bbb Q}$ implies the classical Leopoldt conjecture for a number field $K$ of degree $n$ over ${\Bbb Q}$, if one assumes further the existence of automorphic induction of characters for the extension $K/{\Bbb Q}$. We study Hida's conjecture using the automorphy lifting techniques adapted to the ${\rm GL}_n$ 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 ${\rm GL}_n$ over CM fields without needing to assume automorphic induction of characters for the extension $K/{\Bbb Q}$. For this methods of potential automorphy results are used.