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Mumford-Tate Groups and DomainsTheir Geometry and Arithmetic (AM-183)$
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Mark Green, Phillip A. Griffiths, and Matt Kerr

Print publication date: 2012

Print ISBN-13: 9780691154244

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691154244.001.0001

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(p.1) Introduction
Mumford-Tate Groups and Domains

Mark Green

Phillip Griffiths

Matt Kerr

Princeton University Press

This book deals with Mumford-Tate groups, the fundamental symmetry groups in Hodge theory. Much, if not most, of the use of Mumford-Tate groups has been in the study of polarized Hodge structures of level one and those constructed from this case. In this book, Mumford-Tate groups M will be reductive algebraic groups over ℚ such that the derived or adjoint subgroup of the associated real Lie group M contains a compact maximal torus. In order to keep the statements of the results as simple as possible, the book emphasizes the case when M itself is semi-simple. The discussion covers period domains and Mumford-Tate domains, the Mumford-Tate group of a variation of Hodge structure, Hodge representations and Hodge domains, Hodge structures with complex multiplication, arithmetic aspects of Mumford-Tate domains, classification of Mumford-Tate subdomains, and arithmetic of period maps of geometric origin.

Keywords:   period domain, Mumford-Tate group, Hodge theory, polarized Hodge structure, Lie group, Mumford-Tate domain, Hodge structure, Hodge representation, Hodge domain, complex multiplication

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