Arithmetical Algebraic Geometry

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This award provides funds for studies of modular forms, abelian varieties, and Galois representations. Gerhard Frey's construction, which associates triples of integers to elliptic curves over the rational numbers, has led to a direct link between the investigator's subject and simple problems involving whole numbers. Frey's construction has led to a proof of Fermat's Last Theorem and a resolution of several allied problems in mathematics. Many questions remain to be resolved - the ABC conjecture is perhaps the best known. One expects that arithmetic algebraic geometry will continue to flourish; the investigator is working on some of the problems that are now "on the table" as well as other problems which emerge over time. 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.