Refined Conjectures

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This project supports the research of Professor Tate on 1) various refined conjectures on special values of L-functions, such as Gross's conjectural refined class number formula and the corresponding refinements of Stark's conjectures, 2) questions about the fine structure of the Stark unit in a cyclic extension of a real quadratic field, and 3) Sklyanin algebras of dimension 5. This research lies in the general areas of number and arithmetic geometry. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems. Arithmetic geometry combines two of the oldest branches of mathematics: number theory and geometry. The new insights arising out of this combination are producing increasingly powerful tools to solve long-standing problems like Fermat's Last Theorem, which have resisted the strongest efforts of over three centuries of mathematicians. In addition, though Arithmetic Geometry is sometimes regarded as among the purest of pure mathematics, it has also been developing insightful new techniques leading to dramatic progress in such applied areas as error-correcting codes and cryptography.