Collaborative Research: FRG: Class numbers, hyperbolic manifolds and dynamical systems

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

DMS-0139772 Darren Long The problem of constructing infinitely many number fields of class number one descends from Gauss's work on quadratic forms, and remains one of the major unsolved problems in algebraic number theory. This has been the inspiration for many new ideas and techniques in algebraic number theory and arithmetic geometry. The proposers plan a sustained attack on this problem using geometric methods based upon recent advances in the theory of hyperbolic manifolds coupled with a natural broadening of the classical Bianchi- Hurwitz theorem. Possible important mathematical by-products include new techniques for estimating class numbers and an improvement of the understanding of properties of the trace fields of hyperbolic manifolds. Further consequences of this work of a somewhat broader impact also seem possible. The most obviously relevant applications seem to come from the direction of cryptography. The difficulty of factoring numbers into primes is the basis for the RSA cryptosystem. The fastest known algorithm for factoring large whole numbers into primes is the so-called number field sieve, and the techniques from algebraic number theory involved in this sieve were developed in part from work on class numbers. This project could impact such algorithms and progress on factoring problems would have significant implications for cryptography.