Computing in Jacobians of projective curves over finite fields

Details Computing in Jacobians of projective curves over finite fields Computing in Jacobians of projective curves over finite fields

2010-03-12

arxiv.org/abs/1003.2563v1

Mathematics

Algebraic Geometry

42 pages

Scientific paper

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 many desirable operations can be done 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 Kummer maps; and finding a basis for the $l$-torsion of the Picard group, where $l$ is a prime number different from the characteristic of the base field.

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