Applying Arakelov theory

Abstract

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.