Ariyan Javanpeykar (Universiteit Leiden - Université Paris XI)
Arakelov theory and covers of curves
Let X be a curve over a number field. We study invariants of X such as the Belyi degree and the Faltings height. Our main result is an explicit inequality relating these invariants. As a first application, we deduce a conjecture of Edixhoven-de Jong-Schepers. Secondly, we give an algorithmic application. Finally, we give a Diophantine application: Szpiro's small points conjecture is true for hyperelliptic curves. We finish the talk with a short discussion of Edixhoven's strategy to compute etale cohomology in polynomial time and how the above results fit into this program.