A Hilbert modular surface parametrizes abelian surfaces with real multiplication, and by virtue of this moduli interpretation comes equipped with two natural classes of cycles. The first class of cycles represent abelian surfaces having complex multiplication by a fixed order in a quartic CM field. The second class of cycles are the Hirzebruch-Zagier cycles which represent abelian surfaces having quaternionic multiplication by a fixed order in an indefinite quaternion algebra over the rationals. The cycles of complex multiplication points lie in codimension two, while the Hirzebruch-Zagier cycles lie in codimension one. Hilbert modular surfaces are known to admit integral models which are regular of dimension three, hence one can compute the intersection multiplicity of a codimension two cycle of complex multiplication points with a codimension one Hirzebruch-Zagier cycle. The principal investigator will prove that if the codimension two cycle is held fixed while the Hirzebruch-Zagier cycle varies, then the resulting intersection multiplicities are the Fourier coefficients of a particular weight two modular form. Of particular interest is the case in which the two cycles intersect improperly. This modular form is defined by restricting a parallel weight one Hilbert modular Eisenstein series on a totally real field, viewed as a function on the product of two upper half planes, to the diagonally embedded upper half plane.
Elliptic curves and their higher dimensional variants, called abelian varieties, play an increasingly important role in modern cryptology, and the construction of secure encryption algorithms can sometimes be reduced to the problem of constructing abelian varieties with prescribed properties. The principal investigator will count the number of abelian varieties of dimension two (abelian surfaces) which possess a prescribed collection of symmetries. As the precise collection of symmetries is allowed to vary, the resulting counting formulae are expected to agree with a known sequence of numbers appearing in the theory of modular forms. This latter sequence is readily computable, and thus the resulting formulae will provide an effective method to count the number of abelian surfaces with prescribed properties.
