Siegel modular forms of degree two and three and invariant theory

This is a survey based on the construction of Siegel modular forms of degree 2 and 3 using invariant theory in joint work with Fabien Clery and Carel Faber.

Modular forms of degree two and curves of genus two in characteristic two

We describe the ring of modular forms of degree 2 in characteristic 2 using its relation with curves of genus 2.

The ring of modular forms of degree two in characteristic three

We determine the structure of the ring of Siegel modular forms of degree 2 in characteristic 3.

Covariants of binary sextics and modular forms of degree 2 with character

We use covariants of binary sextics to describe the structure of modules of scalar-valued or vector-valued Siegel modular forms of degree 2 with character, over the ring of scalar-valued Siegel modular forms of even weight. For a modular form defined by a covariant we express the order of vanishing along the locus of products of elliptic curves in terms of the covariant.

On vector-valued Siegel modular forms of degree 2 and weight (j,2)

We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it. We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes.

Covariants of binary sextics and vector-valued Siegel modular forms of genus two

We extend Igusas description of the relation between invariants of binary sextics and Siegel modular forms of degree two to a relation between covariants and vector-valued Siegel modular forms of degree two. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree two.

Exploring modular forms and the cohomology of local systems on moduli spaces by counting points

This is a report on a joint project in experimental mathematics with Jonas Bergstrom and Carel Faber where we obtain information about modular forms by counting curves over finite fields.

A stratification on the moduli of K3 surfaces in positive characteristic

We review the results on the cycle classes of the strata defined by the height and the Artin invariant on the moduli of K3 surfaces in positive characteristic obtained in joint work with Katsura and Ekedahl. In addition we prove a new irreducibility result for these strata.

Divisors on Hurwitz spaces: an appendix to The cycle classes of divisorial Maroni loci

The Maroni stratification on the Hurwitz space of degree $d$ covers of genus $g$ has a stratum that is a divisor only if $d-1$ divides $g$. Here we construct a stratification on the Hurwitz space that is analogous to the Maroni stratification, but has a divisor for all pairs $(d,g)$ with $d leq g$ with a few exceptions and we calculate the divisor class of an extension of these divisors to the compactified Hurwitz space.

The cycle classes of divisorial Maroni loci

We determine the cycle classes of effective divisors in the compactified Hurwitz spaces of curves of genus g with a linear system of degree d that extend the Maroni divisors on the open Hurwitz space. Our approach uses Chern classes associated to a global-to-local evaluation map of a vector bundle over a generic $P^1$-bundle over the Hurwitz space.