Explicit Coleman integration for curves

Abstract

The Coleman integral is a 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 [Tui16, Tui17] to compute the action of Frobenius on 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.