## Computing in Picard groups of projective curves over finite fields

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- by Peter Bruin
- Math. Comp.
**82**(2013), 1711-1756 - DOI: https://doi.org/10.1090/S0025-5718-2012-02650-0
- Published electronically: September 14, 2012
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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.## References

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## Bibliographic Information

**Peter Bruin**- Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich
- Email: peter.bruin@math.uzh.ch
- 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: https://doi.org/10.1090/S0025-5718-2012-02650-0
- MathSciNet review: 3042583