Arithmetic Algebraic Geometry

Princeton University, Princeton NJ

Investigators

Abstract

Abstract for the award DMS-0701395 of Katz The principal investigator proposes to continue work in arithmetic algebraic geometry, especially the l-adic cohomology of varieties over finite fields, exponential sums over finite fields, their associated L-functions, 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, the equidistribution of character sums as the multiplicative character varies, and some 'horizontal' equidistribution questions attached to fixed varieties over Z[1/n] and varying primes p for which no theoretical framework, even conjectural, yet exists. 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 understanding of the analogies between the finite field case and the number field case, analogies which have already played an important role in shaping our very thinking about number theory.