Algorithms for Gromov-Witten Invariants of Elliptic Curves

Abstract: We present an enhanced algorithm for exploring mirror symmetry for elliptic curves through the correspondence of algebraic and tropical geometry, focusing on Gromov-Witten invariants of elliptic curves and, in particular, Hurwitz numbers. We present a new highly efficient algorithm for computing generating series for these numbers. We have implemented the algorithm both using Singular and OSCAR. The implementations outperform by far the current method provided in Singular. The OSCAR implementation, benefiting in particular from just-in-time compilation, again by far outperforms the implementation of the new algorithm in Singular. This advancement in computing the Gromov-Witten invariants facilitates a study of number theoretic and geometric properties of the generating series, including quasi-modularity and homogeneity.