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.