Approximate computations with modular curves

TL;DR

This paper introduces methods for efficient approximate computations with modular curves and their Jacobians, highlighting recent algorithmic advances in algebraic geometry and number theory.

Contribution

It presents new polynomial-time algorithms for approximating modular curves and Jacobians, with applications to number theory.

Findings

Approximate computations are feasible in polynomial time.

Illustrative examples demonstrate the methods.

Applications in number theory are sketched.

Abstract

This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory.