Arithmetical Algebraic Geometry

9700871 Ulmer数论中的一个基本问题是在定义在整体域上的椭圆曲线上构造无穷阶有理点。 迄今为止，最成功的通用方法是使用Heegner点。简而言之，我们通过在上半平面中指定点来定义模曲线上的某些因子;复数乘法理论允许我们证明这些因子是在数域上定义的，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穷阶，这可以通过使用格罗斯和扎吉尔的方法将它们的高度与L函数的特殊值进行比较来完成。第一个项目乌尔默将追求的是扩展这些方法的情况下，椭圆曲线定义的领域的函数曲线在有限领域。 具体而言，他建议开发类似物的结果格罗斯和Zagier有关价值的L-功能的高度特殊点志村曲线。 这将允许一个证明，某些Heegner点，一个先验的理性，有无限的秩序，从而证明了猜想的伯奇和Swinnerton-Dyer椭圆曲线的功能领域的L-功能消失简单。 第二个项目乌尔默将调查也涉及椭圆曲线的功能领域。 与数域情形一样，椭圆曲线上的有理点群也是随机生成的。 另一方面，局部点群非常大--它是一个无限秩的Zp-模。 他建议构建一个子模块的局部点是有限的Zp-秩和包含的全球点。 与Heegner点构造相反，这种方法产生的点是无限阶的先验;在某些情况下，人们可以识别出在基域上的有理点的Z-模，从而提供了无限阶全局点的另一种构造的希望。 第三个项目涉及模p伽罗瓦表示附加到经典的模形式。 具体而言，乌尔默计划使用的存在大量供应的同余之间的模块化形式证明在他以前的工作，以研究几何参数空间的p-adic模块化形式所建造的科尔曼。 他还希望将模表示的一个性质（"扭曲的平凡性“，粗略地说，这是表示在被限制为p处的分解群时可约的条件）与"斜率”联系起来，即，Hecke特征值的估值 这个项目福尔斯属于算术几何的一般领域-一个融合了两个最古老的数学领域：数论和几何的主题。 事实证明，这种结合非常富有成效，最近解决了几代人努力解决的问题。 它的许多后果包括新的纠错码，用于计算机存储设备，如光盘和硬盘驱动器，以及用于互联网上金融交易的安全信息传输方案。

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.