Mathematical Sciences: Zeta Functions and L-Functions over Finite Fields

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9300389 Wan This award supports the research of Professor Wan to work in number theory. A basic question in number theory is to determine the p-adic values of the zeros and poles of zeta functions over finite fields. Professor Wan proposes to give a systematic investigation of this question for families of zeta functions and L-functions. One method is to use Dwork's p-adic theory. A new key ingredient is the introduction of some maximizing functions from linear programming. Professor Wan expects to resolve the Adolphson-Sperber conjecture about the generic Newton polygon of L- functions of exponential sums and prove a strong version of the Dwork-Mazur conjecture about the generic Newton polygon of zeta functions of hypersurfaces. This is research in the field of number theory. Number theory starts with the whole numbers and questions such as the divisibility of one whole number by another. It is among the oldest fields of mathematics and it was originally pursued for purely aesthetic reasons. However, within the last half century, it has become an essential tool in developing new algorithms for computer science and new error correcting codes for electronics. ***