Cohomology of moduli spaces of curves of genus four

Moduli space of curves are one of the central objects of study in modern algebraic geometry and much effort has been devoted to determining their geometry and in particular their cohomology. There has also been significant progress the last few years.

The aim of the present project is to experimentally and theoretically expand our knowledge about the cohomology of moduli spaces of pointed curves of genus four. The algebraic part of the cohomology, and especially its tautological part, is amenable to computations. But the non-algebraic cohomology is much more mysterious.

In this part of the cohomology we expect to find Galois representations directly connected to some automorphic forms. Such connections would be instances of reciprocity in the large mathematical project called the Langlands program.

But if the genus is four or higher, nobody has (to our knowledge) a good idea about what type of automorphic forms these should be. Our study will be based on computer aided counts of curves over finite fields with a small number of elements. This data will be combined with the Lefschetz trace formula to get cohomological information.

We will also use invariant theory to create Teichmüller modular forms, and we will connect them to Galois representations appearing in the cohomology.