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Computing in Picard groups of projective curves over finite fields

Author: Peter Bruin
Journal: Math. Comp. 82 (2013), 1711-1756
MSC (2010): Primary 11G20, 11Y16, 14Q05
Published electronically: September 14, 2012
MathSciNet review: 3042583
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Abstract: 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

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)
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.