Intersections of Hirzebruch-Zagier divisors

希尔伯特模曲面用实乘法参数化了阿贝尔曲面，并且凭借这种模解释，配备了两类自然的循环。第一类环表示四次CM域中具有复乘固定阶的阿贝尔曲面。第二类循环是Hirzebruch-Zagier循环，它表示在无定四元数代数上以固定顺序进行四元数乘法的阿贝尔曲面。复乘法点的循环位于余维二，而Hirzebruch-Zagier循环位于余维一。已知希尔伯特模曲面允许三维正则的积分模型，因此可以计算余维为一的Hirzebruch-Zagier循环的复数乘法点的余维二循环的交多重。首席研究员将证明，如果余维二环保持固定，而Hirzebruch-Zagier环变化，则所得的相交多重度是特定权重两模形式的傅里叶系数。特别令人感兴趣的是这两个周期不恰当相交的情况。这种模形式是通过将一个希尔伯特模爱森斯坦级数在全实数域上的平行权值限定为两个上半平面积上的函数，限定为对角线嵌入的上半平面来定义的。椭圆曲线及其高维变体阿贝尔变体在现代密码学中发挥着越来越重要的作用，安全加密算法的构造有时可以归结为构造具有规定性质的阿贝尔变体问题。首席研究员将计算具有规定的对称性集合的二维（阿贝尔曲面）的阿贝尔变体的数量。由于对称的精确集合是允许变化的，因此所得到的计数公式应该与模形式理论中出现的已知数列一致。后一个序列很容易计算，因此所得公式将提供一种有效的方法来计算具有规定性质的阿贝尔曲面的数量。

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.