A nondefinability result for expansions of the ordered real field by the Weierstrass $\wp$ function

Abstract

Suppose that $\Omega$ is a complex lattice that is closed under complex conjugation and that $I$ is a small real interval, and that $D$ is a disc in $ \mathbb{C}$. Then the restriction $\wp|_D$ is definable in the structure $(\bar{\mathbb{R}},\wp|_I)$ if and only if the lattice $\Omega$ has complex multiplication. This characterises lattices with complex multiplication in terms of definability.