Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects

Reynald Lercier (1) , Christophe Ritzenthaler (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 IML - Institut de mathématiques de Luminy

We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We focus on genus 3 hyperelliptic curves. Both geometric and arithmetic aspects are considered.

Mots clés: Weighted projective space; Reconstruction; Moduli space; Invariant; Algorithm; Binary form; Automorphism; Covariant; Field of moduli; Field of definition; Genus 3

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

Reynald Lercier, Christophe Ritzenthaler. Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects. Journal of Algebra, 2012, 372, pp.595-636. ⟨10.1016/j.jalgebra.2012.07.054⟩. ⟨hal-00694121⟩