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 数论中的一个基本问题是在全局域上定义的椭圆曲线上构造无限阶有理点。 迄今为止，最成功的通用方法是使用 Heegner 点。简而言之，通过指定上半平面中的点来定义模曲线上的某些除数；复数乘法理论允许人们证明这些除数是在数域上定义的，并且使用模参数化可以得到椭圆曲线上的有理点。 然后需要测试这些点是否具有无限阶，这可以通过使用 Gross 和 Zagier 方法将它们的高度与 L 函数的特殊值进行比较来完成。乌尔默将开展的第一个项目是将这些方法扩展到在有限域上的曲线函数域上定义的椭圆曲线的情况。 具体来说，他建议开发 Gross 和 Zagier 结果的类似物，将 L 函数的值与 Shimura 曲线上特殊点的高度相关联。 这将允许人们证明某些先验有理数的 Heegner 点具有无限阶，从而证明 Birch 和 Swinnerton-Dyer 对于 L 函数简单消失的函数域上的椭圆曲线的猜想。 乌尔默将研究的第二个项目也与函数域上的椭圆曲线有关。 与数域情况一样，这种椭圆曲线上的有理点组是有限生成的。 另一方面，局部点群非常大——它是一个无限阶的 Zp 模。 他建议构建一个局部点的子模块，该子模块具有有限的 Zp 秩并包含全局点。 与 Heegner 点构造相反，该方法产生的点是无限阶的先验；在某些情况下，我们可以识别出地面场上推测有理的点的 Z 模，从而为无限阶全局点的另一种构造提供了希望。 第三个项目涉及附加到经典模形式的 mod p Galois 表示。 具体来说，Ulmer 计划利用他之前的工作中证明的模形式之间存在大量同余来研究 Coleman 构造的 p 进模形式参数空间的几何形状。 他还希望将模块化表示的属性（“扭曲的平凡性”，粗略地说，当限制为 p 处的分解群时表示可简化的条件）与“斜率”，即与估值相关联。 Hecke 特征值。 该项目属于算术几何的一般领域——这一学科融合了两个最古老的数学领域：数论和几何。 事实证明，这种结合非常富有成效，最近解决了几代人努力的问题。 其众多后果之一是用于计算机存储设备（如光盘和硬盘驱动器）的新纠错码以及用于互联网上金融交易的安全信息传输方案。