An algebraic model for rational naive-commutative ring SO(2)-spectra and equivariant elliptic cohomology

Equipping a non-equivariant topological $E_\infty$-operad with the trivial $G$-action gives an operad in $G$-spaces. For a $G$-spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups. Algebras over this operad are called na\"{i}ve-commutative ring $G$-spectra. In this paper we take $G=SO(2)$ and we show that commutative algebras in the algebraic model for rational $SO(2)$-spectra model rational na\"{i}ve-commutative ring $SO(2)$-spectra. In particular, this applies to show that the $SO(2)$-equivariant cohomology associated to an elliptic curve $C$ from previous work of the second author is represented by an $E_\infty$-ring spectrum. Moreover, the category of modules over that $E_\infty$-ring spectrum is equivalent to the derived category of sheaves over the elliptic curve $C$ with the Zariski torsion point topology.