Hyperelliptic curves of genus three over finite fields of even characteristic

Authors:
Enric Nart;Daniel Sadornil
Affiliations:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain;Depto. de Algebra y Geometra, Universidad de Valladolid, F. Ciencias, Prado de la Magdalena s/n, 47005 Valladolid, Spain
Venue:
Finite Fields and Their Applications
Year:
2004

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Abstract

In this paper we classify hyperelliptic curves of genus 3 defined over a finite field k of even characteristic. We consider rational models representing all k-isomorphy classes of curves with a given arithmetic structure for the ramification divisor and we find necessary and sufficient conditions for two models of the same type to be k-isomorphic. Also, we compute the automorphism group of each curve and an explicit formula for the total number of curves.