| Modular forms play a central and increasing important role in modern mathematics, in algebraic geometry, number theory and also in string theory. The theory of elliptic modular forms is very well developed, not in the last place because of the abundance of accessible examples. By contrast, for modular forms on higher-dimensional domains such easily accessible examples are lacking. It is the intention of the present proposal to produce many examples of Siegel modular forms, esp. of genus 3 and to come forward with precise conjectures on such Siegel modular forms. We intend to do this by using the relation between modular forms and the cohomology of local systems on the moduli space of abelian varieties of genus 3. This cohomology can be studied by counting curves over finite fields. The plan extends the line of approach started in joint work with Faber and continued in work with Bergstroem for genus 2 Siegel modular forms. We also plan to extend this to other types of modular forms like Picard modular forms. As a byproduct we hope to gain also insight in the cohomology of local systems on the moduli spaces of curves of low genus. The project poses big challenges, both of theoretical as well as of algorithmic and computational nature. |
