Heterogeneous spaces are analogues for curves of genus greater than 1 of principal homogeneous spaces for abelian varieties. Using cocycles on an abelian variety covering the Jacobian, the authors show how to construct a finite number of heterogeneous spaces \(X_ i\) mapped into a curve C, such that the map on their disjoint union is an epimorphism over a field k. The authors illustrate this by solving several higher genus curves, mapping the \(X_ i\) to elliptic curves. They also give explicit formulas for 3-descent of elliptic curves.