Banff International Research Station for Mathematical Innovation and Discovery
Abstract
Humbert surfaces are certain surfaces embedded in the moduli space A2β of principally polarized abelian surfaces. In this talk I will explain the connection between the components of the intersection of two Humbert surfaces and classes of certain binary quadratic forms. More precisely, for each positive quadratic form q in r variables one can associate a closed subvariety H(q) of A2β (which depends only on the equivalence class of the form). If r=1, then we recover the Humbert surfaces. For r=2 we get curves which can be used to describe the intersection of two Humbert surfaces. (Using the reduction theory of binary quadratic forms, this can be done quite explicitly.) If q is a primitive binary quadratic form, then H(q) is irreducible, but in the general case H(q) is a union of the images of modular curves (modular correspondences) lying on X(N)xX(N) (or on Hilbert modular surfaces). By studying conjugacy classes of matrices mod N, the irreducible components of H(q) can be identified. Thus, one gets an explicit description of all irreducible components of the intersection of two Humbert surfaces.Non UBCUnreviewedAuthor affiliation: Queen's UniversityFacult