Super Elliptic Curves

TL;DR

This paper explores super elliptic curves as super Riemann surfaces of genus one, examining their geometric properties, Picard groups, isogenies, and their connection to the super KP hierarchy, revealing new insights into their algebraic and geometric structure.

Contribution

It provides a detailed analysis of super elliptic curves, including their Picard groups, isogenies, and their role in the super KP hierarchy, highlighting differences from classical elliptic curves.

Findings

The Picard group map is a projection, not an isomorphism, for super tori.

The addition law on Pic is realized via intersections with superlines.

Solutions to the super KP hierarchy are constructed from super elliptic curves.

Abstract

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.