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Computational Aspects of Modular Forms and Galois Representations by Robin de Jong, Jean-Marc Couveignes, Franz Merkl, Johan Bosman, Bas Edixhoven

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Chapter Nine

Applying Arakelov theory

by Bas Edixhoven and Robin de Jong

 

In this chapter we start applying Arakelov theory in order to derive a bound for the height of the coefficients of the polynomials Image as in (8.2.10). We proceed in a few steps. The first step, taken in Section 9.1, is to relate the height of the bl(Qx,i) as in Section 8.2 to intersection numbers on Xl. The second step, taken in Section 9.2, 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 ...

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