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

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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.