Abstract. In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field k and the intersection of the moduli space M b 3 of such curves with the hyperelliptic moduli H 3 . Such intersection S is an irreducible, 3-dimensional, rational algebraic variety. We determine the equation of this space in terms of the Gl(2, k)-invariants of binary octavics as defined in [27] and find a birational parametrization of S. We also compute all possible subloci of curves for all possible automorphism group G. Moreover, for every rational moduli point p ∈ S, such that |Aut (p)| > 4, we give explicitly a rational model of the corresponding curve over its field of moduli in terms of the Gl(2, k)-invariants. genus 3 hyperelliptic curves and dihedral invariants and absolute invariants

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It is proved that up to isomorphism there is only one (2, 2)-dimensional supertorus associated to a nontrivial representation of its underlying 2-torus, and that it has nontrivial odd brackets. This supertorus is obtained by finding out first a canonical form for its Lie superalgebra, and then using Lie's technique to represent it faithfully as supervector fields on a supermanifold. Those supervector fields can be integrated, and through their various integral flows the composition law for the supergroup is straightforwardly deduced. It turns out that this supertorus is precisely the supergroup described by Guhr (1993) following a formal analogy with the classical unitary group U(2) but with no further intrinsic characterization.2000 Mathematics Subject Classification: 17B70, 58A50, 58C50, 81R05.

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