Seminar in Algebraic Geometry and Number Theory

Abstract: I will discuss the relation between moduli of two-marked tori and modular forms for a special affine group, which is non-reductive and thus outside of reach of the Langlands Program. The connection is made by the Siegel-Veech transform, which I will explain. In general, the Siegel-Veech transform yields a supply of square-integrable functions on strata of abelian differentials, of which moduli of marked tori are a special case. Before our work, only the case of one-marked tori was understood. In particular, even in the case of two-marked tori spectral properties of the image were unknown. To reveal them we require three differential operators associated with various group actions or foliations. This yields a surprisingly detailed understanding of the L2-space surpassing what we know for the special linear group. On the geometric side, we need to identify novel configurations of relative saddle connections, which I will explain as well. This is based on work with Jayadev Athreya, Jean Lagacé, and Martin Möller.