Elliptic constant cycle curves on Kummer surfaces

The order of a constant cycle curve $C\subset X$ on a K3 surface, defined by Huybrechts, is a positive integer that measures the obstruction to decomposing the diagonal class ${\mathrm{\Delta }}_C$ in the Chow group ${\mathrm{C}\mathrm{H}}^2(X\times C)$. In this paper, we compute the order of elliptic constant cycle curves that naturally arise on Kummer surfaces, by passing to the transcendental intermediate Jacobian $J_{\mathrm{t}\mathrm{r}}^3(X\times C)$. As a consequence, every $n\in \mathbb{N}$ can be realized as the order of a constant cycle curve on a K3 surface.