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GLOBALLY REALIZABLE COMPONENTS OF LOCAL DEFORMATION RINGS

Published online by Cambridge University Press:  03 September 2020

Frank Calegari
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA (fcale@math.uchicago.edu; emerton@math.uchicago.edu)
Matthew Emerton
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA (fcale@math.uchicago.edu; emerton@math.uchicago.edu)
Toby Gee
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK (toby.gee@imperial.ac.uk)

Abstract

Let $n$ be either $2$ or an odd integer greater than $1$, and fix a prime $p>2(n+1)$. Under standard ‘adequate image’ assumptions, we show that the set of components of $n$-dimensional $p$-adic potentially semistable local Galois deformation rings that are seen by potentially automorphic compatible systems of polarizable Galois representations over some CM field is independent of the particular global situation. We also (under the same assumption on $n$) improve on the main potential automorphy result of Barnet-Lamb et al. [Potential automorphy and change of weight, Ann. of Math. (2) 179(2) (2014), 501–609], replacing ‘potentially diagonalizable’ by ‘potentially globally realizable’.

Type
Research Article
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

The first author was supported in part by NSF Grants DMS-1404620 and DMS-1701703. The second author was supported in part by NSF Grants DMS-1303450, DMS-1601871, and DMS-1902307. The third author was supported in part by a Leverhulme Prize, EPSRC Grant EP/L025485/1, Marie Curie Career Integration Grant 303605, ERC Starting Grant 306326, and a Royal Society Wolfson Research Merit Award.

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