Geometric constructions over finite fields, elementary equivalence of finitely generated fields, and rational points on varieties

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

DMS-0301280 Bjorn M. Poonen The investigator proposes: 1) to use sieve methods to prove the existence of various geometric objects and maps over finite fields. For many problems, the standard dimension-based arguments work only over infinite fields; the idea is to replace them by numerical counting. 2) to continue the study initiated by F. Pop on the question of whether elementarily equivalent finitely generated fields are necessarily isomorphic. 3) to write a graduate text highlighting the application of schemes to the study of varieties over nonalgebraically closed fields. Originally, algebraic geometry was concerned with the real number solutions to systems of polynomial equations, but the 20th century made it clear that it was fruitful to generalize by considering fields (i.e., number systems) other than the real numbers. For example, finite fields have proved invaluable in the theory of error-correcting codes and cryptography. Unfortunately, many geometric constructions are known to work only over infinite fields. The investigator's first project is to use combinatorial techniques to adapt these constructions to the finite field case. The investigator's second project concerns the relationships between geometry over different fields; potentially it could allow results over one field to be transferred to another field. Finally, the investigator proposes to write a graduate text that will make newly developed techniques in this area accessible to students.