The geometry of some parameterizations and encodings.

*(English)*Zbl 1314.14053The paper deals with the problem of the parameterization by radicals (of degree prime to \(q(q-1)\)) of algebraic curves of small genus (mainly genus 2) defined over a finite field \(\mathbb{F}_q\), of characteristic different from 2 and 3. A such parameterization would allow an efficient deterministic encodings into those curves.

Encoding algorithms for elliptic curves are already known, see [T. Icart, Lect. Notes Comput. Sci. 5677, 303–316 (2009; Zbl 1252.94075)], but for genus 2 curves only partial results were known, results only valid for a negligible proportion of all genus 2 curves, see Kammerer, Lercier and Renault [J.-G. Kammerer et al., Lect. Notes Comput. Sci. 6487, 278–297 (2010; Zbl 1290.94100)].

Section 2 begins remembering the basic notions of parameterizations, encoding and torsors. Then Tartaglia-Cardan formulae is formulated in the language of torsors and the proposed method of parametrization is showed (2.7 and 2.8). This general method is illustrated in Section 3 for elliptic curves.

Section 4 studies the case of genus 2 curves having two rational points whose difference has order 3 in the associated Jacobian variety. This allows to parameterize by 3-radicals a positive proportion of all genus 2 curves. An example (over \(\mathbb{F}_{83}\)) is worked in detail. Finally the paper studies other parameterizations by \(l\)-radicals, in particular for \(l=5\).

Encoding algorithms for elliptic curves are already known, see [T. Icart, Lect. Notes Comput. Sci. 5677, 303–316 (2009; Zbl 1252.94075)], but for genus 2 curves only partial results were known, results only valid for a negligible proportion of all genus 2 curves, see Kammerer, Lercier and Renault [J.-G. Kammerer et al., Lect. Notes Comput. Sci. 6487, 278–297 (2010; Zbl 1290.94100)].

Section 2 begins remembering the basic notions of parameterizations, encoding and torsors. Then Tartaglia-Cardan formulae is formulated in the language of torsors and the proposed method of parametrization is showed (2.7 and 2.8). This general method is illustrated in Section 3 for elliptic curves.

Section 4 studies the case of genus 2 curves having two rational points whose difference has order 3 in the associated Jacobian variety. This allows to parameterize by 3-radicals a positive proportion of all genus 2 curves. An example (over \(\mathbb{F}_{83}\)) is worked in detail. Finally the paper studies other parameterizations by \(l\)-radicals, in particular for \(l=5\).

Reviewer: Juan Tena Ayuso (Valladolid)

##### MSC:

14H45 | Special algebraic curves and curves of low genus |

14E20 | Coverings in algebraic geometry |

11G20 | Curves over finite and local fields |

12G05 | Galois cohomology |

11S20 | Galois theory |

##### Keywords:

algebraic curves; parameterizations; radicals; finite fields; deterministic algorithms; encodings; torsors##### References:

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