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

合作研究：FRG：类数、双曲流形和动力系统

• 批准号: 0139875
• 负责人: Alan Reid
• 金额: $ 33.26万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139875&HistoricalAwards=false
• 关键词: Collaborative; Research; FRG; Class; numbers

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.

艾伦·里德：第一类无穷多个数域的构造问题源于高斯关于二次型的研究，是代数数论中尚未解决的主要问题之一。这启发了代数数论和算术几何中的许多新思想和新技术。在双曲流形理论最新进展的基础上，结合经典Bianchi- Hurwitz定理的自然展宽，提出了利用几何方法持续解决这一问题的方案。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迹场性质的理解的改进。这项工作的进一步结果似乎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方向。将数分解成质数的困难是RSA密码系统的基础。已知将大整数分解为质数的最快算法是所谓的“数域筛法”（number field sieve），这种筛法中涉及的代数数论技术部分是在对classnumber的研究中发展起来的。这个项目可能会影响这些算法，而分解问题的进展将对密码学产生重大影响。该奖项由拓扑学与代数、数论和组合学项目共同资助。