Users without a subscription are not able to see the full content.

## Bas Edixhoven and Jean-Marc Couveignes

Print publication date: 2011

Print ISBN-13: 9780691142012

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691142012.001.0001

Show Summary Details
Page of

PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2017. 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 http://www.universitypressscholarship.com/page/privacy-policy).date: 12 December 2017

# Short introduction to heights and Arakelov theory

Chapter:
(p.79) Chapter Four Short introduction to heights and Arakelov theory
Source:
Computational Aspects of Modular Forms and Galois Representations
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691142012.003.0004

This chapter discusses bounding the heights of the coefficients of minimal polynomial P. As was hinted at in Chapter 3, such bounds are obtained using Arakelov theory, a tool that is discussed in this chapter. It is not at all excluded that a direct approach to bound the coefficients of P exists, thus avoiding the complicated theory that we use. On the other hand, it is clear that the use of Arakelov theory provides a way to split the work into smaller steps, and that the quantities occurring in each step are intrinsic in the sense that they do not depend on coordinate systems or other choices one could make. This method does not depend on cancellations of terms in the estimates we will do; all contributions encountered can be bounded appropriately.

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.