Counting tropical elliptic plane curves with fixed $j$-invariant

Abstract

In complex algebraic geometry, the problem of enumerating plane elliptic curves of given degree with fixed complex structure has been solved by R. Pandharipande [8] using Gromov–Witten theory. In this article we treat the tropical analogue of this problem, the determination of the number $E_{\mathrm{trop}}(d)$ of tropical elliptic plane curves of degree $d$ and fixed “tropical $j$-invariant” interpolating an appropriate number of points in general position and counted with multiplicities. We show that this number is independent of the position of the points and the value of the $j$-invariant and that it coincides with the number of complex elliptic curves (with $j$-invariant $j\notin \{0,1728\}$). The result can be used to simplify G. Mikhalkin's algorithm to count curves via lattice paths (see [6]) in the case of rational plane curves.