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

DMS-0139772 Darren Long构造第一类无穷多个数域的问题源于Gauss关于二次型的工作，一直是代数数论中尚未解决的主要问题之一。这是代数数论和算术几何中许多新思想和新技术的灵感来源。基于双曲流形理论的最新进展和经典Bianci-Hurwitz定理的自然扩展，提出者计划用几何方法对这个问题进行进一步的攻击。可能的重要数学副产品包括估计类数的新技术和对双曲流形的迹场性质的理解的改进。这项工作的进一步结果也有可能产生更广泛的影响。最明显的相关应用似乎来自密码学的方向。将数字分解成素数的困难是RSA密码系统的基础。将大整数分解成素数最快的算法是所谓的数域筛法，而这个筛子中涉及的代数数论的技术部分是从对类数的工作中发展出来的。这个项目可能会影响这些算法，而因子分解问题的进展将对密码学产生重大影响。

DMS-0139772Darren LongThe 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.