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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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Arithmetic Aspects of Mumford-Tate Domains

Arithmetic Aspects of Mumford-Tate Domains

(p.213) Chapter VI Arithmetic Aspects of Mumford-Tate Domains
Mumford-Tate Groups and Domains

Mark Green

Phillip Griffiths

Matt Kerr

Princeton University Press

This chapter describes the arithmetic aspects of Mumford-Tate domains and Noether-Lefschetz loci. It first clarifies a few points concerning the structure and construction of Mumford-Tate domains before presenting a computationally effective procedure to determine the components in terms of Lie algebra representations and Weyl groups. It then shows that the normalizers of M in G are the groups stabilizing the Noether-Lefschetz locus. It also discusses the decomposition of Noether-Lefschetz loci into Hodge orientations, Weyl groups and permutations of Hodge orientations, and Galois groups and fields of definition. The results demonstrate that Mumford-Tate groups built up from well-understood real² factors are one source of easily described examples of Mumford-Tate domains.

Keywords:   Mumford-Tate domain, Lie algebra representation, Weyl group, Noether-Lefschetz locus, Hodge orientation, Galois group, Mumford-Tate group

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