Functional relations for the order parameters of the chiral Potts model: low-temperature expansions

This is the third in a series of papers in which we set up and discuss the functional relations for the “split rapidity line” correlation function in the N-state chiral Potts model. The order parameters of the model can be obtained from this function. Here we consider the case N=3 and write the equations explicitly in terms of the hyperelliptic functions parametrization. We also present four-term low-temperature series expansions, which we hope will cast light on the analyticity properties needed to solve the relations. The problem remains unsolved, but we hope that this will prove to be a step in the right direction.