The Green's Function on Rhombic Flat Tori

Abstract

We obtain the Green's function $G$ for any flat rhombic torus $T$, always with numerical values of significant digits up to the fourth decimal place (noting that $G$ is unique for $|T|=1$ and $\int_TGdA=0$). This precision is guaranteed by the strategies we adopt, which include theorems such as the Legendre Relation, properties of the Weierstra\ss\,P-Function, and also the algorithmic control of numerical errors. Our code uses complex integration routines developed by H. Karcher, who also introduced the symmetric P-Weierstra\ss\,function, and these resources simplify the computation of elliptic functions considerably.