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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Explicit Coleman integration for curves
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by Jennifer S. Balakrishnan and Jan Tuitman HTML | PDF
Math. Comp. 89 (2020), 2965-2984 Request permission

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.
  • © 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