Intersections of special cycles on Shimura varieties

DMS-0556174 Benjamin Howard主要研究员正在研究Gross-Zagier定理的推广，将Shimura簇上特殊循环的交叉多重性与自守$L$-函数的中心值和中心导数联系起来。 Gross和Zagier的原始定理将模曲线上的复数乘法点的算术交表示为某种显式模形式的傅里叶系数。 格罗斯和扎吉尔利用这一点来推导出模块雅可比矩阵上的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的结果和插图表明，Gross-Zagier定理仅仅是一个更广泛的理论的最简单的情况下，有关特殊圈的算术交叉志村品种的傅立叶系数的模形式。 这样的理论将产生伯奇和斯温纳顿-戴尔猜想的推广形式的结果，例如布洛赫-加藤猜想。为此，首席研究员正在研究Gross-Zagier定理的两种推广形式。 第一个项目是扩展原来的Gross-Zagier定理，包括特殊点的模曲线与额外的水平结构的交点。 这样的结果将产生新的情况下，伯奇和Swinnerton-Dyer猜想的阿贝尔品种附加到模形式与非平凡nevarious nevarious型。 第二个项目是一个庞大的系列的一部分，关于算术交叉的特殊周期的Kudla aptures志村品种的正交型。 感兴趣的情况下，主要研究人员涉及计算交叉多重性一类志村表面，其中包括经典的希尔伯特模块化的表面，并比较这些交叉多重性与傅立叶系数的自守形式。在算术几何领域的某些有一定的曲线，曲面，和高维类似物发挥了核心作用。 这些对象被称为志村变种，并且至少部分地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）而有趣。 这些志村品种包含在他们内部许多有趣的对象较低的维度。 例如，一维的志村变种配备了一族特殊点，二维的志村变种配备了表面上的特殊点和特殊曲线，三维的志村变种内部有特殊点、曲线和表面，等等。这些对象的几何编码算术信息的一种方式是通过相交理论。 例如，如果取一个Shimura曲面和两条位于该曲面上的特殊曲线，则可以简单地计算这两条曲线相交的次数。 Hirzebruch和Zagier的工作可以追溯到20世纪70年代，表明这些几何定义的交叉数与算术中出现的数字序列一致。 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椭圆曲线的基本结果，椭圆曲线在纯数学（例如费马最后定理的证明）和密码学中都具有重要意义。 主要研究人员正在努力将一些理论扩展到更高的维度，通过计算曲面与一系列曲线的交叉数，所有这些都在三维志村品种内，并将这些与算术产生的数字进行比较。 首席研究员期望这将导致一些长期存在的数论重要命题的特殊情况的证明。

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.