# Applying Arakelov theory

# Applying Arakelov theory

This chapter starts applying Arakelov theory in order to derive a bound for the height of the coefficients of the polynomials *P*no hexa conversion found as in (8.2.10). It proceeds in a few steps. The first step is to relate the height of the *b*ₗ(*Q*subscript *x,i*) as in the ptrevious chapter to intersection numbers on *X*ₗ. The second step 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 *X*ₗ. 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

Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.