Intersections of special cycles on Shimura varieties

主要研究者正在研究Gross-Zagier定理的推广，将Shimura变体上特殊循环的交多重性与自同构$L$-函数的中心值和中心导数联系起来。Gross和Zagier的原始定理用某种显式模形式的傅里叶系数来表示模曲线上复数乘法点的算术交。Gross和Zagier用这个推导出了模雅可比矩阵上Heegner点的Neron-Tate高度与l函数的导数之间的关系。Borcherds, Gross-Kudla, Hirzebruch-Zagier， Kudla-Rappoport-Yang和Zhang的结果和猜想表明，Gross-Zagier定理仅仅是一个更广泛的理论的最简单的例子，这个理论将Shimura变体上的特殊循环的算术交与模形式的傅里叶系数联系起来。这样的理论将产生一般化形式的Birch猜想和Swinnerton-Dyer猜想，例如Bloch-Kato猜想。为此，首席研究员正在研究Gross-Zagier定理的两种广义形式。第一个项目是扩展原来的Gross-Zagier定理，使其包含附加水平结构的模曲线上特殊点的交。这样的结果将产生Birch猜想和Swinnerton-Dyer猜想的新情况，这些猜想适用于非平凡nebenttype模形式上的阿贝尔变体。第二个项目是Kudla关于正交型的Shimura变种上的特殊循环的算术相交的一系列猜想的一部分。主要研究者感兴趣的案例涉及计算一类Shimura曲面（包括经典Hilbert模曲面）上的交集多重度，并将这些交集多重度与自同构形式的傅立叶系数进行比较。在算术几何领域中，某些曲线、曲面和高维类似物起着核心作用。这些对象被称为志村变量，它们之所以有趣，至少部分是因为它们以几何形式编码算术信息（即整数和有理数的属性）。志村变体包含了许多有趣的低维物体。例如，一维志村品种具有一系列特殊点，二维志村品种在表面上具有特殊点和特殊曲线，三维志村品种在其内部具有特殊点、曲线和曲面，等等。这些物体的几何形状编码算术信息的一种方法是通过相交理论。例如，如果一个人有一个志村曲面和两条特殊的曲线在曲面上，那么他可以简单地计算两条曲线相交的次数。早在1970年代，Hirzebruch和Zagier的工作就表明，这些几何定义的交点数与算术中产生的数列是一致的。几何和算术之间的这种联系后来被格罗斯和扎吉尔用来证明关于椭圆曲线的基本结果，椭圆曲线在纯数学（例如费马大定理的证明）和密码学中都是非常重要的对象。首席研究员正在通过计算具有一系列曲线的曲面的交点数，将该理论扩展到更高的维度，所有曲线都在三维志村变化中，并将其与算术产生的数字进行比较。首席研究员期望这将导致证明一些长期存在的和重要的数论猜想的特殊情况。

DMS-0556174Benjamin HowardThe principal investigator is studying generalizations of the Gross-Zagier theorem, relating intersection multiplicities of special cycles on Shimura varieties to central values and central derivatives ofautomorphic $L$-functions. The original theorem of Gross and Zagier expresses the arithmetic intersections of complex multiplication points on modular curves in terms of Fourier coefficients of a certain explicit modular form. Gross and Zagier use this to deduce a relation between the Neron-Tate heights of Heegner points on modular Jacobians and derivatives of L-functions. Results and conjectures of Borcherds, Gross-Kudla, Hirzebruch-Zagier, Kudla-Rappoport-Yang, and Zhang suggest that the Gross-Zagier theorem is merely the simplest cases of a much broader theory relating arithmetic intersections of special cycles on Shimura varieties to Fourier coefficients of modular forms. Such a theory would yield results toward generalized forms of the Birch and Swinnerton-Dyer conjecture, e.g. the Bloch-Kato conjectures. Toward this end, the principal investigator is studying two generalized forms of the Gross-Zagier theorem. The first project is to extend the original Gross-Zagier theorem to include intersections of special points on modular curves with additional level structure. Such a result would yield new cases of the Birch and Swinnerton-Dyer conjecture for abelian varieties attached to modular forms with nontrivial nebentype. The second project is a part of a vast series of conjectures of Kudla concerning the arithmetic intersections of special cycles on Shimura varieties of orthogonal type. The case of interest to the principal investigator involves the computation of intersection multiplicities on a class of Shimura surfaces which includes the classical Hilbert modular surfaces, and the comparison of these intersection multiplicities with Fourier coefficients of automorphic forms.In the field of arithmetic geometry certain there are certain curves, surfaces, and higher dimension analogs which play a central role. These objects are called Shimura varieties, and are interesting at least in part because they encode arithmetic information (i.e. properties of the integers and rational numbers) in a geometric form. These Shimura varieties contain inside them many interesting objects of lower dimensions. For example the one-dimensional Shimura varieties come equipped with a family of special points, the two-dimensional Shimura varieties come equipped with both special points and special curves on the surface, three-dimensional Shimura varieties have special points, curves, and surfaces inside them, and so on. One way in which the geometry of these objects encodes arithmetic information is through ntersection theory. If, for example, one takes a Shimura surface and two special curves lying on the surface, then one may simply count the number of times that the two curves intersect one another. Work of Hirzebruch and Zagier, dating back to the 1970's, shows that these geometrically defined intersection numbers agree with sequences of numbers arising in arithmetic. This connection between geometry and arithmetic was later exploited by Gross and Zagier to prove fundamental results about elliptic curves, objects of great importance both in pure math (e.g. to the proof of Fermat's last theorem) and in cryptography. The principal investigator is working to extend some of the theory to higher dimensions by computing the intersection numbers of a surface with a family of curves, all inside of a three-dimensional Shimura variety, and comparing these with numbers arising from arithmetic. The principal investigator expects that this will lead to proofs of special cases of some long-standing and important conjectures in number theory.