On the spectral curve of the difference Lamé operator

Abstract

We give two "complementary" descriptions of the curve $Γ$ parametrizing double-Bloch solutions to the difference analogue of the Lamé equation. The curve depends on a positive integer number $\ell$ and two continuous parameters: the "lattice spacing" $η$ and the modular parameter $τ$. Apart from being a covering of the elliptic curve with the modular parameter $τ$, $Γ$ is a hyperelliptic curve of genus $2\ell$. We also point out connections between the spectral curve and representations of the Sklyanin algebra.