You can find a fully worked-out derivation of the propagator on an elliptic curve in Appendix A of <A HREF="http://arxiv.org/abs/1112.4015">Feynman Graph Integrals and Almost Modular Forms</A> by S. Li (2011). Basically, the propagator $P(z)$ appears upon evaluation of $\sum_{n,m=-\infty}^{\infty}(z-m-n)^{-2}$.