Anogia Summer school, July 23–29, 2005

Number fields and curves over finite fields

Preliminary Program:

• Robin de Jong: In this lecture series we will discuss the basic notions and ideas of Arakelov theory for curves over a number field. We will see how Arakelov theory gives rise to some interesting arithmetic invariants. We will study some explicit examples of these invariants and we we will show how certain bounds for these invariants imply the Mordell conjecture. Literature.
• Michael Tsfasman: The title is Asymptotic theory of global fields. We will relate the contents of the papers Infinite global fields and the generalized Brauer-Siegel theorem (pdf) and Asymptotic behaviour of the Euler-Kronecker constant (pdf)
• Gerard van der Geer: Curves over finite fields and modular forms. In these lectures we discuss curves and abelian varieties over finite fields. After discussing the basics we intend to show how counting curves over finite fields can be used to obtain information about modular forms.
• René Schoof: In these lectures we present classical algebraic number theory from the point of view of Arakelov theory. We discuss the relations between Arakelov divisors and ideal lattices and applications to fast algorithms to compute Arakelov class groups (Buchmann's algorithm, infrastructure). See [1], [2].