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

TL;DR

This paper proves that the theory of the structure of complex numbers with CM points is stable and decidable, countering previous suggestions of its undecidability, using an effective version of the André-Oort conjecture.

Contribution

It demonstrates the decidability and stability of the theory of complex numbers with CM points using an effective André-Oort conjecture, challenging prior assumptions.

Findings

The structure of CM points in complex numbers is decidable.

The theory of this structure is stable.

Counterexample to the conjecture of undecidability of $ ext{Th}( ext{CM})$.

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.