Arithmetical Algebraic Geometry

算术代数几何

• 批准号: 9700871
• 负责人: Douglas Ulmer
• 金额: $ 7.5万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9700871&HistoricalAwards=false
• 关键词: Arithmetical; Algebraic; Geometry

9700871 Ulmer A fundamental problem in number theory is to construct rational points of infinite order on elliptic curves defined over global fields. To date, the most successful general method has used Heegner points. Briefly, one defines certain divisors on a modular curve by specifying points in the upper half-plane; the theory of complex multiplication allows one to show that these divisors are defined over a number field, and using a modular parameterization one gets rational points on the elliptic curve. One needs then to test whether these points have infinite order, which can be done by comparing their heights with special values of L-functions using the method of Gross and Zagier. The first project Ulmer will pursue is extending these methods to the case of elliptic curves defined over the fields of functions of curves over finite fields. Specifically, he proposes to develop the analogues of the results of Gross and Zagier relating values of L-functions to heights of special points on Shimura curves. This will allow one to prove that certain Heegner points, a priori rational, have infinite order and thereby prove the conjecture of Birch and Swinnerton-Dyer for elliptic curves over function fields whose L-function vanishes simply. The second project Ulmer will investigate is also related to elliptic curves over function fields. As in the number field case, the group of rational points on such an elliptic curve is finitely generated. On the other hand, the group of local points is very big--it is a Zp-module of infinite rank. He proposes to construct a submodule of the local points which is of finite Zp-rank and which contains the global points. In contrast to the Heegner point construction, this method yields points which are a priori of infinite order; in some cases one can identify a Z-module of points which are conjecturally rational over the ground field, thus offering the hope of an alternative construction of global points of infinite order. The third project deals with the mod p Galois representations attached to classical modular forms. Specifically, Ulmer plans to use the existence of a large supply of congruences between modular forms proved in his previous work to study the geometry of a parameter space of p-adic modular forms constructed by Coleman. He also hopes to relate a property of modular representations (``twisted ordinarity'', which is roughly speaking the condition that the representation be reducible when restricted to a decomposition group at p) to ``slopes'', i.e., to the valuations of Hecke eigenvalues. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful, having recently solved problems that withstood the efforts of generations. Among its many consequences are new error correcting codes which are used in computer storage devices like compact disks and hard drives and secure information transmission schemes which are used for financial transactions on the internet.

9700871 Ulmer数字理论的一个基本问题是在全球领域定义的椭圆曲线上构建无限顺序的合理点。 迄今为止，最成功的一般方法已经使用了Heegner点。简而言之，一个人通过指定上半平面中的点来定义模块化曲线上的某些除数；复杂乘法的理论允许人们证明这些除数是在数字字段上定义的，并且使用模块化参数化，一个在椭圆曲线上获得有理点。 然后需要测试这些点是否具有无限顺序，可以通过使用总和和Zagier的方法将其高度与特殊值的高度进行比较。乌尔默（Ulmer）将追求的第一个项目是将这些方法扩展到在有限场上曲线功能场上定义的椭圆曲线的情况。 具体而言，他建议开发总和Zagier的结果，将L功能的值与Shimura曲线上的特殊点的高度相关。 这将使人们可以证明某些Heegner点具有无限顺序，从而证明了Birch和Swinnerton-Dyer的猜想，用于椭圆形曲线，而椭圆曲线的L功能仅消失了。 第二个项目Ulmer将研究还与功能场上的椭圆曲线有关。 与数字场案例一样，这种椭圆曲线上的理性点组有限地生成。 另一方面，本地点非常大 - 它是无限等级的ZP模块。 他建议构建有限的ZP级别的当地点的子模块，其中包含全球要点。 与Heegner Point构造相反，该方法产生的点是无限顺序的先验点；在某些情况下，人们可以识别一个Z模块，这些点在地面上是有理由的，因此可以替代全球无限秩序点的替代构造。 第三个项目涉及附加在经典模块化表单上的Mod P Galois表示形式。 具体而言，乌尔默（Ulmer）计划利用他以前的工作中证明的模块化形式之间的大量一致性，以研究由科尔曼（Coleman）构建的P-ADIC模块化形式的参数空间的几何形状。 他还希望将模块化表示的属性（``扭曲的顺式''联系起来，该属性大致说明，当将表示形式限制为p）时可以还原为``斜坡''，即``斜坡''，即``'''，即即hecke eigenvalues的估值。 该项目属于算术几何形状的一般领域 - 一个融合了数学最古老的领域的主题：数字理论和几何学。 事实证明，这种组合非常富有成果，最近解决了与世代相传的努力的问题。 它的许多后果包括新的错误纠正代码，这些代码用于计算机存储设备（如紧凑型磁盘和硬盘驱动器）以及安全的信息传输方案，这些方案用于Internet上的金融交易。