Ben Howard (Boston)

Hirzebruch-Zagier divisors and CM cycles on Hilbert modular surfaces

Conjectures of Kudla predict that the intersection multiplicities of special cycles on the integral models of certain Shimura varieties should be related to the Fourier coefficients of automorphic forms. As a particular example, one can take the Shimura variety to be a Hilbert modular surface X. The integral model of X is then an arithmetic threefold which comes equipped with two different types of special cycles. First one has the locus of points on X which admit complex multiplication by the maximal order in a quartic CM field, and this locus is a cycle of codimension two. Second one has the locus of points on X which admit an action by an order in a quaternion algebra. As the quaternionic order varies one obtains the family of Hirzebruch-Zagier divisors. We will show that the intersection multiplicities of the CM cycles with the Hirzebruch- Zagier divisors agree with the Fourier coefficients of a very particular modular form, which arises as the central derivative of a Hilbert modular Eisenstein series. This is joint work with Tonghai Yang.