Jump to ContentJump to Main Navigation
Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)$
Users without a subscription are not able to see the full content.

Umberto Zannier

Print publication date: 2012

Print ISBN-13: 9780691153704

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153704.001.0001

Show Summary Details
Page of

PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2018. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in HSO for personal use (for details see www.princeton.universitypressscholarship.com/page/privacy-policy).date: 18 October 2018

About the André-Oort Conjecture

About the André-Oort Conjecture

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

Umberto Zannier

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153704.003.0005

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

Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.