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

点击翻译按钮获取中文摘要

DMS-0139875Alan ReidThe problem of constructing infinitely many number fields of classnumber one descends from Gauss's work on quadratic forms, and remainsone of the major unsolved problems in algebraic number theory. Thishas been the inspiration for many new ideas and techniques inalgebraic number theory and arithmetic geometry. The proposers plan asustained attack on this problem using geometric methods based uponrecent advances in the theory of hyperbolic manifolds coupled with anatural broadening of the classical Bianchi- Hurwitz theorem.Possible important mathematical by-products include new techniquesfor estimating class numbers and an improvement of theunderstanding of properties of the trace fields of hyperbolicmanifolds.Further consequences of this work of a somewhat broader impact alsoseem possible. The most obviously relevant applications seem tocome from the direction of cryptography. The difficulty of factoringnumbers into primes is the basis for the RSA cryptosystem. The fastestknown algorithm for factoring large whole numbers into primes is theso-called number field sieve, and the techniques from algebraic numbertheory involved in this sieve were developed in part from work on classnumbers. This project could impact such algorithms and progress onfactoring problems would have significant implications for cryptography. This award is jointly funded by the programs in Topology and Algebra,Number Theory, & Combinatorics.