SUPER ELLIPTIC-CURVES

A detailed study is made of super elliptic curves, namely super Rieman n surfaces of genus one considered as algebraic varieties, particularl y their relation with their Picard groups. This is the simplest settin g in which to study the geometric consequences of the fact that certai n cohomology groups of super Riemann surfaces with odd spin structure are not freely generated modules. The divisor theory of Rosly, Schwarz , and Voronov gives a map from a supertorus to its Picard group Pic, b ut this map is a projection, not an isomorphism as it is for ordinary tori. The geometric realization of the addition law on Pic via interse ctions of the supertorus with superlines in projective space is descri bed. The isomorphisms of Pic with the Jacobian and the divisor class g roup are verified. All possible isogenies, or surjective holomorphic m aps between supertori, are determined and shown to induce homomorphism s of the Picard groups. Finally, the solutions to the new super Kadomt sev-Petviashvili hierarchy of Mulase-Rabin which arise from super elli ptic curves via the Krichever construction are exhibited.