## Explicit Coleman integration for curves

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Jennifer S. Balakrishnan and Jan Tuitman
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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.## References

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## Additional Information

**Jennifer S. Balakrishnan**- Affiliation: Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, Massachusetts 02215
- MR Author ID: 910890
- Email: jbala@bu.edu
**Jan Tuitman**- Affiliation: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
- MR Author ID: 941045
- Email: jan_tuitman@hotmail.com
- Received by editor(s): November 14, 2017
- Received by editor(s) in revised form: January 10, 2020, January 16, 2020, and March 2, 2020
- Published electronically: May 22, 2020
- Additional Notes: The first author was supported in part by NSF grant DMS-1702196, the Clare Boothe Luce Professorship (Henry Luce Foundation), and Simons Foundation grant #550023.

The second author is a Postdoctoral Researcher of the Fund for Scientific Research FWO - Vlaanderen. - © Copyright 2020 American Mathematical Society
- Journal: Math. Comp.
**89**(2020), 2965-2984 - MSC (2010): Primary 11S80; Secondary 11Y35, 11Y50
- DOI: https://doi.org/10.1090/mcom/3542
- MathSciNet review: 4136553