Studies in arithmetic algebraic geometry

Princeton University, Princeton NJ

Investigators

Abstract

Technical description of the project: The project is concerned with A) the study of algebraic varieties over finite fields, exponential sums on such varieties, their associated L-functions, how these objects vary in families, and the determination of the associated monodromy groups which govern this variation. B)Various "horizontal" questions where we start with a situation over the integers, reduce modulo various primes p, and ask how the answers to the questions of section A) above vary with p. C) Various questions of Lang-Trotter type about families, other than modular families, of elliptic curves in a given characteristic p. Some of the main technical tools for attacking questions of type A) are group theory, Fourier Transform, and the theory of perverse sheaves. For questions of types B) and C), there are very few tools, and even designing and implementing numerical experiments to guess what should be true can be nontrivial. Broader significance and importance of the project: The broader significance and importance of the project is three-fold. While it is too soon to appraise the 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. The project proposes to extend our understanding of already posed, extremely interesting mathematical questions over finite fields, questions the answers to which may well in the future have a broader societal impact. Second, the project proposes the investigation of new, extremely interesting, mathematical questions, for whose consideration there does not yet exist even a theoretical framework; any progress on such questions may point the way to the presently missing theoretical framework. Third, the project will deepen our understanding of the analogies between the finite eld case and the number field case, analogies which have already played a important role in shaping our very thinking about the number field case.