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高斯对二次型的研究，是代数数论中一个主要的未解决问题。 代数数论和算术几何中的许多新思想和新技术都受到了这种启发。 提议者计划使用几何方法对这个问题进行持续的攻击，这些方法基于双曲流形理论的最新进展，加上经典比安奇-赫尔维茨定理的自然扩展。可能的重要数学-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。这项工作的更广泛影响的进一步结果也似乎可能最明显的相关应用似乎来自密码学方向。 将数字分解成素数的困难性是RSA密码系统的基础。将大整数分解为素数的已知最快算法是所谓的数域筛，而涉及到这个筛的代数数论技术部分是从对类数的研究中发展起来的。 这个项目可能会影响这样的算法和进展onfactoring问题将有重大影响的密码学。

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.