The arithmetic of Jacobian groups of superelliptic cubics

International audienceWe 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

On automorphisms of algebraic curves

An irreducible, algebraic curve $\mathcal X_g$ of genus $g\geq 2$ defined over an algebraically closed field $k$ of characteristic \mbox{char } \, k = p \geq 0, has finite automorphism group \mbox{Aut} (\mathcal X_g). In this paper we describe methods of determining the list of groups \mbox{Aut} (\mathcal X_g) for a fixed $g\geq 2$. Moreover, equations of the corresponding families of curves are given when possible