Arithmetical Algebraic Geometry

9700871 Ulmer数论中的一个基本问题是构造定义在整体域上的椭圆曲线上的无限级有理点。到目前为止，最成功的一般方法是使用Heegner点。简单地说，通过指定上半平面上的点来定义模曲线上的某些因子；复数乘法理论允许人们证明这些因子定义在数域上，并且使用模参数化得到椭圆曲线上的有理点。然后，人们需要检验这些点是否具有无穷级，这可以通过使用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实现。Ulmer要追求的第一个项目是将这些方法推广到定义在有限域上的曲线函数域上的椭圆曲线的情况。具体地说，他建议推广Gross和Zagier关于L函数的值与Shimura曲线上特殊点的高度的结果的类比。这将使我们能够证明某些先验有理的Heegner点是无穷级的，从而证明了Birch和Swinnerton-Dyer关于其L函数简单消失的函数域上的椭圆曲线的猜想。Ulmer将要研究的第二个项目也与函数域上的椭圆曲线有关。与数域的情况一样，这种椭圆曲线上的有理点群是有限生成的。另一方面，局部点群是非常大的--它是一个无限秩ZP-模。他建议构造一个局部点的子模，它具有有限的ZP-秩且包含全局点。与Heegner点构造不同的是，这种方法产生的点是无穷级先验的；在某些情况下，人们可以识别在地面上猜想有理的点的Z-模，从而提供了一种替代无穷级全局点的构造的希望。第三个项目研究附加到经典模形式上的mod p Galois表示。具体地说，Ulmer计划利用他以前的工作中证明的模形式之间存在大量同余的性质来研究由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(“扭转正规性”，粗略地说，这是指当限制到p处的分解群时表示是可约的)到“斜率”，即与Hecke特征值的赋值有关。这个项目属于算术几何的一般领域--一个融合了数论和几何这两个最古老的数学领域的学科。事实证明，这种结合非常有成效，最近解决了经得起几代人努力的问题。在其众多后果中，包括用于光盘和硬盘等计算机存储设备的新纠错码，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.