Mathematical Sciences: RUI: Arithmetic of Modular Forms and Algebraic Varieties

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Professor Gouvea will work on several problems connected with the theory of p-adic modular forms. In particular, he will study conjectures on the existence of modular forms connected by congruence conditions. He will also work on the arithmetic properties of elliptic curves in twisted families. 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.