The arithmetic of Jacobian groups of superelliptic cubics

Abdolali Basiri 1, 2 Andreas Enge 3, 4 Jean-Charles Faugère 1, 2 Nicolas Gürel 3, 4
1 SPACES - Solving problems through algebraic computation and efficient software
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
3 TANC - Algorithmic number theory for cryptology
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
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.
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Abdolali Basiri, Andreas Enge, Jean-Charles Faugère, Nicolas Gürel. The arithmetic of Jacobian groups of superelliptic cubics. Mathematics of Computation, American Mathematical Society, 2005, 74 (249), pp.389-410. ⟨inria-00071967v2⟩

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