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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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Computing in Picard groups of projective curves over finite fields
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by Peter Bruin PDF
Math. Comp. 82 (2013), 1711-1756 Request permission


We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that various desirable operations can be performed efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite morphisms, and hence Picard and Albanese maps on Jacobians; generating uniformly random divisors and points on Jacobians; computing Frobenius maps; and finding a basis for the $l$-torsion of the Picard group for prime numbers $l$ different from the characteristic of the base field.
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Additional Information
  • Peter Bruin
  • Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich
  • Email:
  • Received by editor(s): February 4, 2011
  • Received by editor(s) in revised form: November 2, 2011
  • Published electronically: September 14, 2012
  • Additional Notes: This paper evolved from one of the chapters of the author’s thesis [Modular curves, Arakelov theory, algorithmic applications, Proefschrift, Universiteit Leiden, 2010], the research for which was supported by the Netherlands Organisation for Scientific Research (NWO)
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 82 (2013), 1711-1756
  • MSC (2010): Primary 11G20, 11Y16, 14Q05
  • DOI:
  • MathSciNet review: 3042583