L-Functions of Algebraic Varieties Over Finite Fields

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Abstract Daquing Wan UC Irvine 9970417 Professor Wan will continue his study of the p-adic of L-functions arising from algebraic geometry over a finite field of characteristic p. The central problem in this study is a 1973 conjecture of Dwork about the p-adic continuation of the L function of a unit root F-chrystal. Prof. Wan will try to establish the general validity of this conjecture. He will also investigate the arithmetic consequences of the conjecture including higher dimensional generalizations of the Gouvea-Mazur conjecture, and the zeros of certain geometric L-functions, and the connections with the mysterious mirror map that arises from a family of Calabi-Yau varieties. This is a project in number theory and algebraic geometry. One of mathematics oldest branches, number theory is the study of numbers that can be expressed exactly in terms of the simplest counting numbers, 1, 2, 3, 4, 5. In the second half of this century, modern geometry has had a tremendous impact on number theory introducing new methods and new problems. The development of algebraic geometry has paralleled advances in number theory for the past 30 years. Professor Wan will study geometric objects defined over the simplest possible mathematical basis for such a geometry, a finite field. These are simultaneously the simplest and most intriguing objects in algebraic geometry. They have found very practical application in communications, security, and other surprising areas.