Galois Representations and (Phi, Gamma)-Modules
Galois Representations and (Phi, Gamma)-Modules

Author: Peter Schneider

Publisher: Cambridge University Press

Published: 2017-04-20

Total Pages: 157

ISBN-13: 110718858X

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A detailed and self-contained introduction to a key part of local number theory, ideal for graduate students and researchers.

Automorphic Forms and Galois Representations: Volume 1
Automorphic Forms and Galois Representations: Volume 1

Author: Fred Diamond

Publisher: Cambridge University Press

Published: 2014-10-16

Total Pages: 385

ISBN-13: 1316062333

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Automorphic forms and Galois representations have played a central role in the development of modern number theory, with the former coming to prominence via the celebrated Langlands program and Wiles' proof of Fermat's Last Theorem. This two-volume collection arose from the 94th LMS-EPSRC Durham Symposium on 'Automorphic Forms and Galois Representations' in July 2011, the aim of which was to explore recent developments in this area. The expository articles and research papers across the two volumes reflect recent interest in p-adic methods in number theory and representation theory, as well as recent progress on topics from anabelian geometry to p-adic Hodge theory and the Langlands program. The topics covered in volume one include the Shafarevich Conjecture, effective local Langlands correspondence, p-adic L-functions, the fundamental lemma, and other topics of contemporary interest.

Moduli Stacks of Étale (φ, Γ)-Modules and the Existence of Crystalline Lifts
Moduli Stacks of Étale (φ, Γ)-Modules and the Existence of Crystalline Lifts

Author: Matthew Emerton

Publisher: Princeton University Press

Published: 2022-12-13

Total Pages: 312

ISBN-13: 069124135X

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"Motivated by the p-adic Langlands program, this book constructs stacks that algebraize Mazur's formal deformation rings of local Galois representations. More precisely, it constructs Noetherian formal algebraic stacks over Spf Zp that parameterize étale ([phi], [Gamma])-modules; the formal completions of these stacks at points in their special fibres recover the universal deformation rings of local Galois representations. Matthew Emerton and Toby Gee use these stacks to show that all mod p representations of the absolute Galois group of a p-adic local field lift to characteristic zero, and indeed admit crystalline lifts. They explicitly describe the irreducible components of the underlying reduced substacks and discuss the relationship between the geometry of these stacks and the Breuil-Mézard conjecture. Along the way, they prove a number of foundational results in p-adic Hodge theory that may be of independent interest"--

Automorphic Forms and Galois Representations
Automorphic Forms and Galois Representations

Author: Fred Diamond

Publisher: Cambridge University Press

Published: 2014-10-16

Total Pages: 387

ISBN-13: 1107693632

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Part two of a two-volume collection exploring recent developments in number theory related to automorphic forms and Galois representations.

Automorphic Forms and Galois Representations: Volume 2
Automorphic Forms and Galois Representations: Volume 2

Author: Fred Diamond

Publisher: Cambridge University Press

Published: 2014-10-16

Total Pages: 387

ISBN-13: 1316062341

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Automorphic forms and Galois representations have played a central role in the development of modern number theory, with the former coming to prominence via the celebrated Langlands program and Wiles' proof of Fermat's Last Theorem. This two-volume collection arose from the 94th LMS-EPSRC Durham Symposium on 'Automorphic Forms and Galois Representations' in July 2011, the aim of which was to explore recent developments in this area. The expository articles and research papers across the two volumes reflect recent interest in p-adic methods in number theory and representation theory, as well as recent progress on topics from anabelian geometry to p-adic Hodge theory and the Langlands program. The topics covered in volume two include curves and vector bundles in p-adic Hodge theory, associators, Shimura varieties, the birational section conjecture, and other topics of contemporary interest.

p-adic Differential Equations
p-adic Differential Equations

Author: Kiran S. Kedlaya

Publisher: Cambridge University Press

Published: 2010-06-10

Total Pages: 399

ISBN-13: 1139489208

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Over the last 50 years the theory of p-adic differential equations has grown into an active area of research in its own right, and has important applications to number theory and to computer science. This book, the first comprehensive and unified introduction to the subject, improves and simplifies existing results as well as including original material. Based on a course given by the author at MIT, this modern treatment is accessible to graduate students and researchers. Exercises are included at the end of each chapter to help the reader review the material, and the author also provides detailed references to the literature to aid further study.