Universiteit Leiden Mathematisch Instituut
Geometry Seminar

Wednesday, 15 January 2003, 16:00 - 17:00, room 403

Wednesday, 22 January 2003, 16:00 - 17:00, room 403

Gabor Wiese: Calculating characteristic p Katz modular forms of weight 1, after Bas Edixhoven

Abstract:

The aim of these talks is to present a theorem by Bas Edixhoven, which describes the space of cuspidal Katz modular forms of weight 1 with a given character over a finite field just in terms of the Hecke algebra of classical modular forms (over the complex numbers, for the group Gamma_1(N), without a character).

Moreover, effective bounds on the number of Hecke operators needed are given. Using standard software tools, the calculations can be performed using only linear algebra methods.

We shall also present a variant of the theorem (valid in certain cases) for the calculation of cuspidal Katz modular forms of weight 1 with trivial character, where one uses the classical Hecke algebra for modular forms for Gamma_0(N).

Along the way we introduce Hecke operators, parabolic cohomology and also a way to calculate classical modular forms of weight k > 1.