Explicit Coleman integration for curves

Abstract

The Coleman integral is a p p -adic line integral that plays a key role in computing several important invariants in arithmetic geometry. We give an algorithm for explicit Coleman integration on curves, using the algorithms of the second author [Math. Comp. 85 (2016), pp. 961–981] and [Finite Fields Appl. 45 (2019), pp. 301–322] to compute the action of Frobenius on p p -adic cohomology. We present a collection of examples computed with our implementation. This includes integrals on a genus 55 curve, where other methods do not currently seem practical.