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BIRS Workshop Lecture Videos

Intersections of Humbert surfaces and binary quadratic forms Kani, Ernst


Humbert surfaces are certain surfaces embedded in the moduli space $A_2$ 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 $A_2$ (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 $X(N) x X(N)$ (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.

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