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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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Applying Arakelov theory

Applying Arakelov theory

(p.187) Chapter Nine Applying Arakelov theory
Computational Aspects of Modular Forms and Galois Representations

Bas Edixhoven

Robin de Jong

Princeton University Press

This chapter starts applying Arakelov theory in order to derive a bound for the height of the coefficients of the polynomials Pno hexa conversion found as in (8.2.10). It proceeds in a few steps. The first step is to relate the height of the bₗ(Qsubscript x,i) as in the ptrevious chapter to intersection numbers on Xₗ. The second step is to get some control on the difference of the divisors D₀ and Dₓ as in (3.4). Certain intersection numbers concerning this difference are bounded in Theorem 9.2.5, in terms of a number of invariants in the Arakelov theory on modular curves Xₗ. These invariants will then be bounded in terms of l in Chapter 11. The chapter formulates the most important results, Theorem 9.1.3, Theorem 9.2.1, and Theorem 9.2.5, in the context of curves over number fields, that is, outside the context of modular curves.

Keywords:   modular forms, Arakelov theory, bounding heights, polynomials

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