Publications:

• A Database of Invariant Rings (with Gregor Kemper, Elmar Körding, Gunter Malle, B. Heinrich Matzat, Denis Vogel), (Experimental Mathematics 10 (2001), 537-542). [Preprint version].

  We announce the creation of a database of invariant rings. This database contains a large number of invariant rings of finite groups, mostly in the modular case. It gives information on generators and structural properties of the invariant rings. The main purpose is to provide a tool for researchers in invariant theory.

• Dihedral Galois representations and Katz modular forms (DOCUMENTA MATHEMATICA, Vol. 9 (2004), 123-133)

  We show that any two-dimensional odd dihedral representation \rho over a finite field of characteristic p>0 of the absolute Galois group of the rational numbers can be obtained from a Katz modular form of level N, character \epsilon and weight k, where N is the conductor, \epsilon is the prime-to-p part of the determinant and k is the so-called minimal weight of \rho. In particular, k=1 if and only if \rho is unramified at p. Direct arguments are used in the exceptional cases, where general results on weight and level lowering are not available.

• Computing Hecke algebras of weight 1 in MAGMA, Appendix B of the article by Bas Edixhoven entitled Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one (arXiv:math.NT/0312019), Journal of the Inst. of Math. Jussieu (2006) 5(1), 1-34. [Preprint version of the appendix].

  We report on an implementation in Magma of the algorithm for calculating Hecke algebras of Katz modular forms exposed in the main article. Moreover, results of some computations are included.

• On the faithfulness of parabolic cohomology as a Hecke module over a finite field. Journal für die reine und angewandte Mathematik 606 (2007), 79-103. Preprint version on the arXiv.

  In this article we prove that under certain conditions the Hecke algebra of cuspidal modular forms over F_p coincides with the Hecke algebra of a certain parabolic group cohomology group with coefficients in F_p. These results can e.g. be used to compute Katz modular forms of weight one over an algebraic closure of F_p with methods of linear algebra over F_p.

• On modular symbols and the cohomology of Hecke triangle surfaces. Accepted for publication in International Journal of Number Theory.

  The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surfaces and group cohomology. In all the note a general commutative ring is used as coefficient ring in view of applications to the computation of modular forms over rings different from the complex numbers.

• On the failure of the Gorenstein property for Hecke algebras of prime weight (with L.J.P. Kilford). Experimental Mathematics 17(1), 2008, 37-52.

  In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that the attached mod p Galois representation is unramified at p and the Frobenius at p acts by scalars. The results lead us to the ask the question whether the Gorenstein defect and the multplicity of the attached Galois representation are always equal to 2. We review the literature on the failure of the Gorenstein property and multiplicity one, discuss in some detail a very important practical improvement of the modular symbols algorithm over finite fields and include precise statements on the relationship between the Gorenstein defect and the multiplicity of Galois representations. Appendix A: Manual of Magma package HeckeAlgebra, Appendix B: Tables of Hecke algebras.

• Multiplicities of Galois representations of weight one (with an appendix by Niko Naumann). Algebra and Number Theory, 1:1 (2007), 67-85. Preprint version on arXiv.

  In this article we consider mod p modular Galois representations which are unramified at p such that the Frobenius element at p acts through a scalar matrix. The principal result states that the multiplicity of any such representation is bigger than 1.

Last modification: 2 June 2008.