You can find a fully worked-out derivation of the propagator on an elliptic curve in Appendix A of Feynman Graph Integrals and Almost Modular Forms by S. Li (2011). Basically, the propagator $P(z)$ appears upon evaluation of $\sum_{n,m=-\infty}^{\infty}(z-m-n)^{-2}$.
Carlo Beenakker
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