FQHE and Jain's approach on the torus

By using the explicit knowledge of the lowest energy single particle wave functions in the presence of an {\it arbitrary} magnetic field, we extend to the case of a torus Jain's idea of looking at the FQHE as a manifestation of an integer effect for {\it composite fermions}. We show that this can be realized thanks to a redefinition of the vacuum state that is explicitly collective in nature. We also discuss the relationship of this approach with the hierarchical scheme and with the characterization of the Hall states in terms of $W_{1+\infty}$ algebras and 2D conformal field theories.