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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 of Period Maps of Geometric Origin

Arithmetic of Period Maps of Geometric Origin

(p.269) Chapter VIII Arithmetic of Period Maps of Geometric Origin
Mumford-Tate Groups and Domains

Mark Green

Phillip Griffiths

Matt Kerr

Princeton University Press

This chapter considers some arithmetic aspects of period maps with a geometric origin. It focuses on the situation Φ‎ : S(ℂ) → Γ‎\D, where S parametrizes a family XS of smooth, projective varieties defined over a number field k. The chapter recalls the notion of absolute Hodge classes (AH) and strongly absolute Hodge classes (SAH). The particular case when the Noether-Lefschetz locus consists of isolated points is alluded to in the discussion of complex multiplication Hodge structures (CM Hodge structures). A related observation is that one may formulate a variant of the “Grothendieck conjecture” in the setting of period maps and period domains. The chapter also describes a behavior of fields of definition under the period map, along with the existence and density of CM points in a motivic variation of Hodge structure.

Keywords:   period map, absolute Hodge class, Noether-Lefschetz locus, complex multiplication Hodge structure, Grothendieck conjecture, period domain, Hodge structure

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