  A detailed study is made of super elliptic curves, namely super Riemann
surfaces of genus one considered as algebraic varieties, particularly their
relation with their Picard groups. This is the simplest setting in which to
study the geometric consequences of the fact that certain cohomology groups of
super Riemann surfaces are not freely generated modules. The divisor theory of
Rosly, Schwarz, and Voronov gives a map from a supertorus to its Picard group,
but this map is a projection, not an isomorphism as it is for ordinary tori.
The geometric realization of the addition law on Pic via intersections of the
supertorus with superlines in projective space is described. The isomorphisms
of Pic with the Jacobian and the divisor class group are verified. All possible
isogenies, or surjective holomorphic maps between supertori, are determined and
shown to induce homomorphisms of the Picard groups. Finally, the solutions to
the new super Kadomtsev-Petviashvili (super KP) hierarchy of Mulase-Rabin which
arise from super elliptic curves via the Krichever construction are exhibited.
