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Computational Aspects of Modular Forms and Galois RepresentationsHow One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)$
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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

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Short introduction to heights and Arakelov theory

Short introduction to heights and Arakelov theory

(p.79) Chapter Four Short introduction to heights and Arakelov theory
Computational Aspects of Modular Forms and Galois Representations

Bas Edixhoven

Robin de Jong

Princeton University Press

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.

Keywords:   modular forms, minimal polynomial, Arakelov theory, height functions, arithmetic surfaces

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