Computational Methods in Arithmetic Geometry

这个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重大开放问题。同样的工具也可以用来帮助建立更快、更安全的密码系统，并构建纠错码，使数据能够在不可靠的网络上可靠地传输。初步结果表明，算法的性能和精度应该可以提高几个数量级，这将使解决一些目前无法解决的问题成为可能。该项目的主要目标是实际实现用于计算算术方案的Zeta函数的平均多项式时间算法，特别关注代数曲线。这些算法允许人们有效地计算固定算术方案模所有素数直到指定界的约简的Zeta函数。这正是研究算术统计中的各种问题，并将相关的L函数逼近到高精度所需的计算。这些工具将被用来研究佐藤-塔特分布和朗兰兹计划的明确方面。该项目的计算结果将在开放获取的电子数据库中公布，例如LMFDB(L-功能和模块化表格数据库)。

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).