Explicit Coleman integration for curves

Balakrishnan, Jennifer; Tuitman, Jan

doi:10.1090/mcom/3542

Explicit Coleman integration for curves
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by Jennifer S. Balakrishnan and Jan Tuitman;

Math. Comp. 89 (2020), 2965-2984

DOI: https://doi.org/10.1090/mcom/3542

Published electronically: May 22, 2020

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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 [Math. Comp. 85 (2016), pp. 961–981] and [Finite Fields Appl. 45 (2019), pp. 301–322] 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.

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