Modular Forms of Genus Three

The Eichler-Shimura theorem expresses elliptic modular forms in terms of cohomology of local systems. This theorem generalizes to higher genera. The cohomology of local systems can be explored by counting points over finite fields. We show how this can be applied succesfully to obtain a lot of information about modular forms of genus 2 and 3. This is joint work with Jonas Bergstroem and Carel Faber.