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Mathematics of Computation

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Explicit Coleman integration for curves


Authors: Jennifer S. Balakrishnan and Jan Tuitman
Journal: Math. Comp. 89 (2020), 2965-2984
MSC (2010): Primary 11S80; Secondary 11Y35, 11Y50
DOI: https://doi.org/10.1090/mcom/3542
Published electronically: May 22, 2020
MathSciNet review: 4136553
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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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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.
Article copyright: © Copyright 2020 American Mathematical Society