Computational Methods in Arithmetic Geometry

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

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).