## The arithmetic of Jacobian groups of superelliptic cubics

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- by Abdolali Basiri, Andreas Enge, Jean-Charles Faugère and Nicolas Gürel PDF
- Math. Comp.
**74**(2005), 389-410

## Abstract:

We present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity. The first algorithm, inspired by Cantor’s reduction for hyperelliptic curves, is easily implemented with a few lines of code, making use of a polynomial arithmetic package. We prove explicit reducedness criteria for superelliptic curves of genus 3 and 4, which show the correctness of the algorithm. The second approach, quite general in nature and applicable to further classes of curves, uses the FGLM algorithm for switching between Gröbner bases for different orderings. Carrying out the computations symbolically, we obtain explicit reduction formulae in terms of the input data.## References

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## Additional Information

**Abdolali Basiri**- Affiliation: Laboratoire d’Informatique de Paris 6 (CNRS/UMR 7606), 4 place Jussieu, 75252 Paris Cedex 05, France
- Address at time of publication: Department of Mathematics and Computer Sciences, Damghan University of Sciences, Damghan, Iran
- Email: basiriab2@yahoo.com
**Andreas Enge**- Affiliation: INRIA Futurs and Laboratoire d’Informatique (CNRS/FRE 2653), École polytechnique, 91128 Palaiseau Cedex, France
- Email: enge@lix.polytechnique.fr
**Jean-Charles Faugère**- Affiliation: Laboratoire d’Informatique de Paris 6 (CNRS/UMR 7606), 4 place Jussieu, 75252 Paris Cedex 05, France
- Email: jcf@calfor.lip6.fr
**Nicolas Gürel**- Affiliation: INRIA Futurs and Laboratoire d’Informatique (CNRS/FRE 2653), École polytechnique, 91128 Palaiseau Cedex, France
- Email: gurel@lix.polytechnique.fr
- Received by editor(s): July 18, 2002
- Received by editor(s) in revised form: January 17, 2003
- Published electronically: July 20, 2004
- © Copyright 2004 by the authors
- Journal: Math. Comp.
**74**(2005), 389-410 - MSC (2000): Primary 11G20, 14Q05, 14H40, 14H45, 68W30; Secondary 11T71, 13P10
- DOI: https://doi.org/10.1090/S0025-5718-04-01699-0
- MathSciNet review: 2085899