Computational Methods in Arithmetic Geometry

算术几何的计算方法

• 批准号: 1522526
• 负责人: Andrew Sutherland
• 金额: $ 18万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522526&HistoricalAwards=false
• 关键词: Computational; Methods; Arithmetic; Geometry

This project will develop new computational tools that will allow mathematicians to investigate some major open questions in number theory. The same tools can also be used to help build faster and more secure cryptographic systems, and to construct error-correcting codes that allow data to be reliably transmitted over unreliable networks. Preliminary results indicate that it should be possible to improve the performance and precision of the algorithms involved by several orders of magnitude, which would make it feasible to address a number of questions that are currently out of reach.The bulk of the project will be aimed at the practical realization of average polynomial-time algorithms for computing zeta functions of arithmetic schemes, with a particular focus on algebraic curves. These algorithms allow one to efficiently compute the zeta function of the reduction of a fixed arithmetic scheme modulo all primes up to specified bound. This is precisely the computation needed to investigate various questions in arithmetic statistics, and to approximate the associated L-function to high precision. These tools will be used to study Sato-Tate distributions and explicit aspects of the Langlands program. Computational results of the project will be published in open access electronic databases such as the LMFDB (L-functions and Modular Forms Database).

该项目将开发新的计算工具，使数学家能够研究数论中的一些主要开放问题。同样的工具也可以用来帮助构建更快、更安全的加密系统，并构造纠错码，使数据能够在不可靠的网络上可靠地传输。初步结果表明，应该有可能将所涉算法的性能和精度提高几个数量级，这将使解决目前无法解决的一些问题成为可能，该项目的大部分将着眼于实际实现计算算术方案的zeta函数的平均多项式时间算法，特别侧重于代数曲线。 这些算法允许一个有效地计算zeta函数的减少一个固定的算术计划模所有素数到指定的界限。这正是研究算术统计中的各种问题所需的计算，并将相关的L函数近似到高精度。这些工具将被用来研究Sato-Tate分布和朗兰兹计划的明确方面。该项目的计算结果将在LMFDB（L函数和模块化形式数据库）等开放式电子数据库中发布。