Coleman Integration on Modular Curves

Abstract

Coleman integrals is a major tool in the explicit arithmetic of algebraic varieties, notably in the study of rational points on curves. One of the inputs to compute Coleman integrals is the availability of an affine model. We develop a model-free algorithm that computes single Coleman integrals between any two points on modular curves. Using Hecke operators, any Coleman integral can be broken down into a sum of tiny integrals. We illustrate this using several examples computed in SageMath and Magma. We also suggest some future directions for this work, including a possible extension to iterated Coleman integrals.