*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 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 *b*_{l}(*Q*_{x,i}) as in Section 8.2 to intersection numbers on *X*_{l}. The second step, taken in Section 9.2, is to get some control on the difference of the divisors *D*_{0} and *D*_{x} 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 ...