Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Potential automorphy and the Leopoldt conjecture   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.

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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: solv-int
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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph