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Journal Articles Mathematics of Computation Year : 2005

The arithmetic of Jacobian groups of superelliptic cubics

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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Other [cs.OH]
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Dates and versions

inria-00071967 , version 1 (23-05-2006)
inria-00071967 , version 2 (30-10-2009)

Identifiers

  • HAL Id : inria-00071967 , version 2

Cite

Abdolali Basiri, Andreas Enge, Jean-Charles Faugère, Nicolas Gürel. The arithmetic of Jacobian groups of superelliptic cubics. Mathematics of Computation, 2005, 74 (249), pp.389-410. ⟨inria-00071967v2⟩
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