Explicit Coleman integration for curves

TL;DR

This paper introduces an explicit algorithm for Coleman integration on algebraic curves, leveraging Frobenius action on p-adic cohomology, enabling computations on high-genus curves where previous methods are impractical.

Contribution

It presents a new algorithm for explicit Coleman integration on curves, utilizing Frobenius action, and demonstrates its effectiveness through computational examples including high-genus curves.

Findings

Successful computation of integrals on a genus 55 curve

Implementation handles cases where other methods are impractical

Provides a practical tool for arithmetic geometry computations

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 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.