p-adic Cohomology and Applications

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

Abstract for the award of Kiran Kedlaya DMS-0400272 p-adic analysis, initiated by Hensel at the turn of the last century, seeks to bring the "continuous" techniques of calculus to bear on "discrete" problems in number theory. We study applications of p-adic analysis in arithmetic algebraic geometry; a typical problem is to count solutions of systems of polynomial equations. We work on one hand on improving our theoretical understanding of this problem, and on the other hand on developing practical algorithms for treating important special cases. Some of these cases occur in applications to cryptography, error correcting codes, and other areas of computer science. More specifically, we are developing Berthelot's rigid cohomology in parallel with the older theory of etale cohomology, which is better understood theoretically but ill-equipped for explicit computations. One long-term goal is to extend Lafforgue's work on the function field Langlands correspondence to p-adic sheaves (crystals). In another direction, we are investigating algorithms for computing in p-adic cohomology, which may have mathematical applications on top of the practical ones mentioned above.