PhD thesis: Modular Forms of Weight One Over Finite Fields

The thesis consists of a couple of different projects, all centred around modular forms over finite fields with a special interest in those of weight one.

My main motivation for my work is the desire to understand (a bit) better the absolute Galois group of the rationals. Its 1-dimenstional representations are fully described by class field theory, so that those of dimension 2 are the next natural step. According to a theorem by Deligne Hecke eigenforms mod p give rise to 2-dimensional odd Galois representations over an algebraic closure of F_p. Serre's conjecture proposes a converse: every odd irreducible Galois representation over an algebraic closure of F_p should come from a modular form.

Concerning Serre's conjecture, I managed to prove it in its strictest form (with the minimal weight defined by Edixhoven) for dihedral Galois representations, including, in particular, the exceptional cases when level and weight lowering are not known. This is the content of Chapter 1 and my Documenta article (to publications). Moreover, I performed computer calculations verifying the conjecture in some cases. Apart from that, explicit computations of mod 2 modular forms have yielded the realisation of the group SL_2(F_2^r) for r < 78 as Galois group over the rationals (more information). These and some more computations are reported upon in Chapter 5.

Another connected theme is the study of some Hecke modules with a special regard as to their faithfulness. Natural examples are the cohomology of subgroups of SL_2(Z), the cohomology of modular curves considered as Riemann surfaces, and modular symbols. I gave a concise algebraic treatment of these modules over arbitrary rings, compared them, and gave criteria for their equality in Chapter 2 and the preprint On modular symbols and the cohomology of Hecke triangle surfaces. My principal theoretical contribution concerns the faithfulness of these modules for the weight p if the base ring is F_p after localisation at a prime of the Hecke algebra corresponding to an ordinary modular form. Weight p is the smallest weight that cannot in general be treated any more by p-adic Hodge theory. This is the contents of Chapter III (and parts of Chapter IV), as well as of my article On the faithfulness of parabolic cohomology as a Hecke module over a finite field.

These faithful Hecke modules allow to compute mod p modular forms using the modular symbols formalism over F_p, hence fast methods of linear algebra over a finite field. This formalism has been implemented in MAGMA by William Stein. My theoretical progress enables one to compute mod p modular forms of weight one with mod p modular symbols, using ideas of Edixhoven, as described for instance in my appendix. Earlier, it had been necessary to use characteristic zero. I implemented these algorithms in MAGMA (more information) and described them in Chapter 4.

Last modification: 27 March 2006.