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Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)$
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Umberto Zannier

Print publication date: 2012

Print ISBN-13: 9780691153704

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153704.001.0001

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About the André-Oort Conjecture

About the André-Oort Conjecture

(p.96) Chapter 4 About the André-Oort Conjecture
Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)

Umberto Zannier

Princeton University Press

This chapter discusses some aspects of the so-called André–Oort conjecture. It observes how this resembles the pattern of the statements discussed so far, recalling also a brief history. After a few general considerations about the shape of the André–Oort conjecture and its relation to the topics of the previous chapters, the chapter discusses in some detail what can probably be considered the most basic case of the conjecture, which is distinctly simpler to state. After a section devoted to a review of the basic definitions and results in the context of modular curves and singular invariants, it presents André's original proof. Subsequently it also outlines Pila's proof of the same statement, which follows the method already considered in Chapter 3. The final section discusses in a bit more detail the general definition of Shimura varieties, which appear in the very statements of André–Oort type.

Keywords:   André–Oort conjecture, modular curves, singular invariants, Pila, Shimura varieties

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