Universiteit Leiden Mathematisch Instituut
Geometry Seminar

Wednesday, 29 January 2003, 16:00 - 17:00, room 403

Gabor Wiese: Calculations of characteristic p Katz modular forms of weight 1 and relation to Serre's conjecture

Abstract:

Very vaguely speaking, we will try to outline links - partly conjectural - between the theory of modular forms and number theory, more precisely the study of the absolute Galois group of the rational numbers.

The starting point is a theorem by Deligne, which associates to a Katz eigenform over an algebraic closure of F_p a 2-dimensional odd Galois representation over the same field. Serre conjectured that there is a converse to Deligne's result: given an odd irreducible Galois representation, there exists an eigenform of a precisely specified level, weight and character giving rise to this representation. In particular, the weight is conjectured to be 1 if and only if the representation is unramified at p.

Serre's conjecture has spectacular consequences, among them a direct proof of Fermat's last theorem, which we will sketch.

We will present some of the known results, focussing on dihedral representations.

Finally, we will discuss some calculations giving evidence for Serre's conjecture, which were originally carried out by Mestre, and which we verified.