L-Functions and Monodromy

Princeton University, Princeton NJ

Investigators

Abstract

Abstract for award DMS-0355496 of Katz The principal investigator proposes to continue work in arithmetic algebraic geometry, especially the l-adic cohomology of varieties over finite fields, the l-adic theory of exponential sums over finite fields, the locations of the "low-lying zeroes" of L-functions over finite fields, the determination of monodromy groups, and the application of that determination to the earlier questions. Some of the main tools are group theory, Fourier Transform, and the theory of perverse sheaves. Particular topics of investigation include the calculation of monodromy groups, questions about the L-functions of elliptic curves, and the equidistribution of character sums as the character varies. The broader impact of this project is three-fold. While it is too soon to appraise the wide societal impact of this particular project, the last two decades have seen stunning practical application in many fields (e.g. telecommunications, cryptology, and computer security, to name just a few) of a great deal of algebraic geometry over finite fields, some of which goes back to the nineteenth century, and all of which seemed quite arcane at the time it was being done. On a more immediate scale, the project will lead to a great deal of interaction with postdoctoral fellows, graduate students, and advanced undergraduates, both in theoretical collaborations and in the carrying out of computer experiments. From the narrowest point of view, the project will advance our knowledge in a vital area of mathematics.