Decidability of some complicated structures definable in $\mathbb{C}(t)$

Abstract

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.