# Commentarii Mathematici Helvetici

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**Volume 84, Issue 2, 2009, pp. 387–427**

**DOI: 10.4171/CMH/166**

Published online: 2009-06-30

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

^{[1]}and Hannah Markwig

^{[2]}(1) Universität Kaiserslautern, Germany

(2) Universität des Saarlandes, Saarbrücken, Germany

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`_{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` ∉ { 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.

*Keywords: *Enumerative geometry, tropical geometry

Kerber Michael, Markwig Hannah: Counting tropical elliptic plane curves with fixed `j`-invariant. *Comment. Math. Helv.* 84 (2009), 387-427. doi: 10.4171/CMH/166