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

Suppose that $\mathrm{\Omega }$ 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 $\mathrm{\wp }{\mathrm{\mid }}_D$ is definable in the structure $(\bar{\mathbb{R}},\mathrm{\wp }{\mathrm{\mid }}_I)$ if and only if the lattice $\mathrm{\Omega }$ has complex multiplication. This characterises lattices with complex multiplication in terms of definability.