Intersections of special cycles on Shimura varieties

志村品种特殊周期的交点

• 批准号: 0556174
• 负责人: Benjamin Howard
• 金额: $ 8.57万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0556174&HistoricalAwards=false
• 关键词: Intersections; special; cycles; Shimura; varieties

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.

DMS-0556174Benjamin Howardth The主要研究者正在研究Gross-Zagier定理的概括，将Shimura品种的特殊周期的相交多重性与副型$ l $功能的中心价值和中心衍生物相关。 Gross和Zagier的原始定理以某种显式模块化形式的傅立叶系数表示模块化曲线上复杂乘法点的算术相交。 GROSS和Zagier使用它来推断模块化雅各布人的Neron-Tate高度与L功能的衍生物之间的关系。 Borcherds，Gross-Kudla，Hirzebruch-Zagier，Kudla-Rappoport-Yang和Zhang的结果和猜想表明，Gross-Zagier定理只是一个更广泛的理论的最简单案例，它与Shimura colecles of Shimura Arithmetic互动的更广泛的理论是shimura coledies coledies to for touriel coildier sodiel copefferial copefferial copefferial copefferial sodield sodield sodiquilts oferiel copeffer的形式。 这样的理论将带来桦木和swinnerton-dyer猜想的广义形式的结果，例如Bloch-Kato的猜想。为此，主要研究者正在研究两种总体Zagier定理的广义形式。 第一个项目是将原始的总Zagier定理扩展到模块化曲线上的特殊点的交叉点以及其他级别结构。 这样的结果将产生新的桦木和swinnerton-dyer猜想的新病例，这些猜想是与具有非平凡性nebentype的模块化形式相关的阿贝尔品种。 第二个项目是关于库德拉（Kudla）的大量猜想的一部分，这些库德拉（Kudla）关于正交类型的shimura品种的特殊周期的算术相交。 主要研究者感兴趣的情况涉及在一类Shimura表面上的相交倍数的计算，其中包括经典的希尔伯特模块化表面，以及这些相交倍数的比较与自动形式的傅立叶系数的傅立叶系数。在某些算术几何范围内，某些曲面的竞争范围是某些曲面，并且具有某些曲面的作用。 这些对象称为shimura品种，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）。 这些Shimura品种包含许多较低维度的有趣对象。 例如，一维的Shimura品种配备了一个特殊点的家族，二维Shimura品种配备了表面上的特殊点和特殊曲线，三维Shimura品种具有特殊点，曲线，曲线和表面，内部表面等等。 这些对象的几何形状编码算术信息的一种方法是通过差异理论。 例如，如果一个人采用shimura表面和两条特殊曲线，则可以简单地计算这两条曲线相互相交的次数。 Hirzebruch和Zagier的作品可以追溯到1970年代，表明这些几何定义的相交数与算术中产生的数字序列一致。 Gross和Zagier随后利用了几何形状和算术之间的这种联系，以证明有关椭圆曲线的基本结果，纯数学中非常重要的对象（例如，在Fermat的最后一个定理证明）和密码学中。 主要研究者正在努力通过计算与曲线家族的表面的交点数量，将某些理论扩展到更高的维度，所有这些理论都在三维shimura品种的内部，并将其与算术产生的数字进行比较。 首席研究人员预计，这将导致一些长期和重要猜想的特殊案例的证据。