Schottky problem, Siegel modular forms and quadratic forms

The Schottky problem is about understanding when a principally polarised abelian variety is the Jacobian of a curve. There is a classical way to approach this problem using modular forms. More generally, there is a very rich interplay between modular forms, moduli space of curves and moduli space of abelian varieties. We are particularly interested in theta series, these are modular forms associated to convenient quadratic forms. By proving some geometrical results about the singularities of the Satake compactification of the moduli space of curves, we will show that any given linear combination of theta series is not identically zero on the moduli space of curves of high enough genus; this can be rephrased by saying that there are no stable solutions to the Schottky problem. We are able to make this result effective just for quadratic forms of rank at most 24. This is a joint work with N. Shepherd-Barron. We will also report about a work in progress on the connection between conformal vertex algebras and modular forms defined on the moduli space of curves.