Several properly countable unions of algebraic sets in $\mathbb{ C}^n$ are definable in $\mathbb{ C}(t)$ including the set CM of $j$-invariants of complex elliptic curves with complex multiplication. It has been suggested that one could prove the undecidability of $\operatorname{ Th}(\mathbb{ C}(t))$ by showing that the theory of the structure $\mathsf{ CM} := (\mathbb{ C},+,\cdot,0,1,CM)$ of the field of complex numbers considered with a unary predicate picking out CM is undecidable. We show using an effective version of the Andr\'e-Oort conjecture that to the contrary $\operatorname{ Th}(\mathsf{ CM})$ is stable and decidable. We discuss some related structures on the complex numbers definable in $\mathbb{ C}(t)$ and how their theories may be connected to the Zilber-Pink conjectures.
Categories: math.LO math.AG math.NT