Applications of p-adic Uniformization to Vanishing Cycles and Galois Representations

Bruce Jordan提出研究p进均匀化在算术中的应用。这一主题贯穿于过去十年的一些伟大成就之中，但它现在正被孤立起来，其作用得到承认。例如，它是Ribet证明Serre猜想Epsilon的关键，也是Taylor和Wiles随后证明半稳定椭圆曲线的Shimura-Taniyama猜想的关键成分。对于有理数的算术（这是Ribet和Taylor/Wiles结果的设置），关键是由Cerednik/Drinfeld Shimura曲线是p- ady均匀化的，这使得人们能够同时研究Hecke结构和惯性对雅可比矩阵上有限阶点的质数化简的作用。这里所证明的各种定理都可以推广到任何具有类似p进一致化的情况，甚至可以推广到高维的情况。Jordan想以复杂度递增的顺序来研究这些情况：全实数域上的Shimura曲线，全实数域上的四元Hilbert模变，最后是由酉群统一的Shimura变。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，这一组合非常富有成效——最近解决了几代人经受住了考验的问题。其诸多后果之一是新的纠错码。这些代码对现代计算机（硬盘）和光盘都是必不可少的。

Jordan 9703820 Bruce Jordan proposes to study applications of p-adic uniformization to arithmetic. This theme runs throughout some of the great accomplishments of the last decade, but it is just now being isolated and its role recognized. As one example, it is the key to Ribet's proof of Serre's Conjecture Epsilon and a crucial ingredient to Taylor and Wiles's subsequent proof of the Shimura-Taniyama Conjecture for semi-stable elliptic curves. For arithmetic over the rational numbers (which is the setting for the Ribet and Taylor/Wiles result), the point is that by Cerednik/Drinfeld Shimura curves are p-adically uniformized and this enables one to simultaneously study both the Hecke structure and the action of inertia for a prime of bad reduction on the points of finite order on their jacobians. The various theorems proved here should all generalize to any setting where one has an analogous p-adic uniformization, even to cases of higher dimension. Jordan wants to study these cases in increasing order of complexity: Shimura curves over totally real fields, quaternionic Hilbert modular varieties over totally real fields, and finally Shimura varieties uniformized by unitary groups. 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 generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.