Potential automorphy and the Leopoldt conjecture
American Journal of Mathematics
Johns Hopkins University Press
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Khare, C., & Thorne, J. (2017). Potential automorphy and the Leopoldt conjecture. American Journal of Mathematics, 139 (5), 1205-1273. https://doi.org/10.1353/ajm.2017.0030
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 F = Q implies the classical Leopoldt conjecture for a number field K of degree n over Q, if one assumes further the existence of automorphic induction of characters for the extension K=Q. 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=Q. For this methods of potential automorphy results are used.
In the period during which this research was conducted, Jack Thorne served as a Clay Research Fellow. Chandrashekhar Khare was supported by NSF grant DMS-1161671 and by a Humboldt Research Award, and thanks the Tata Instititute of Fundamental Research, Mumbai for hospitality during the period in which most of the work on this paper was done.
External DOI: https://doi.org/10.1353/ajm.2017.0030
This record's URL: https://www.repository.cam.ac.uk/handle/1810/254249