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

约旦9703820 布鲁斯乔丹建议研究p-adic均匀化在算术中的应用。 这一主题贯穿了过去十年的一些伟大成就，但它只是刚刚被孤立出来，其作用才得到承认。 作为一个例子，它是里贝特证明塞尔猜想的关键，也是泰勒和怀尔斯后来证明志村-谷山猜想的关键。 对于有理数上的算术（这是Ribet和Taylor/Wiles结果的设置），关键是Cerednik/Drinfeld Shimura曲线是p-adically均匀化的，这使得人们能够同时研究Hecke结构和对Jacobian上有限阶点的不良约化素数的惯性作用。 这里证明的各种定理都应该推广到任何具有类似p-adic单值化的设置，甚至更高维的情况。 乔丹想研究这些情况下，增加秩序的复杂性：志村曲线完全真实的领域，四元数希尔伯特模品种完全真实的领域，最后志村品种统一的酉群。 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--最近解决了几代人都无法解决的问题。 在它的许多后果是新的纠错码。 此类代码对于现代计算机（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.