Arithmetical Algebraic Geometry

[700871]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，最成功的一般方法是使用Heegner点。简单地说，通过在上半平面上指定点来定义模曲线上的某些除数；复乘法理论允许人们证明这些除数是在一个数字域上定义的，并且使用模参数化可以在椭圆曲线上得到有理点。然后需要检验这些点是否具有无限阶，这可以通过用Gross和Zagier的方法将它们的高度与l函数的特殊值进行比较来完成。Ulmer的第一个项目是将这些方法扩展到有限域上曲线函数域上定义的椭圆曲线的情况。具体来说，他建议发展Gross和Zagier关于l函数值与Shimura曲线上特殊点高度的结果的类似物。这将允许人们证明某些Heegner点，一个先验有理点，具有无限阶，从而证明Birch和Swinnerton-Dyer关于函数场上l函数简单消失的椭圆曲线的猜想。Ulmer将研究的第二个项目也与函数场上的椭圆曲线有关。在数域的情况下，这样的椭圆曲线上的有理点群是有限生成的。另一方面，局部点的群非常大——它是一个无穷阶的zp模。他提出构造局部点的一个子模，该子模具有有限的zp秩，并且包含全局点。与Heegner点构造相反，这种方法产生的点是无限阶的先验点；在某些情况下，人们可以在地面场上识别出推测有理的点的z模，从而提供了无限顺序的全局点的替代结构的希望。第三个项目处理附加在经典模形式上的模p伽罗瓦表示。具体来说，Ulmer计划利用在他之前的工作中证明的模形式之间的大量同余的存在性来研究Coleman构造的p进模形式的参数空间的几何。他还希望将模表示的一个性质（“扭曲平凡”，粗略地说，它是表示在p处被限制到分解群时可约的条件）与“斜率”联系起来，即与Hecke特征值的赋值联系起来。这个项目属于算术几何的一般领域，它融合了数学中两个最古老的领域：数论和几何。事实证明，这种结合非常富有成效，最近解决了经受几代人努力的问题。它的诸多后果包括用于计算机存储设备（如光盘和硬盘驱动器）的新的纠错码，以及用于互联网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.