Definability and decidability for rings of integers in totally imaginary fields

We show that the ring of integers of ${\mathbb{Q}}^{\text{tr}}$ is existentially definable in the ring of integers of ${\mathbb{Q}}^{\text{tr}}(i)$, where ${\mathbb{Q}}^{\text{tr}}$ denotes the field of all totally real numbers. This implies that the ring of integers of ${\mathbb{Q}}^{\text{tr}}(i)$ is undecidable and first-order non-definable in ${\mathbb{Q}}^{\text{tr}}(i)$. More generally, when $L$ is a totally imaginary quadratic extension of a totally real field $K$, we use the unit groups $R^{\times }$ of orders $R\subseteq {\mathcal{O}}_L$ to produce existentially definable totally real subsets $X\subseteq {\mathcal{O}}_L$. Under certain conditions on $K$, including the so-called JR-number of ${\mathcal{O}}_K$ being the minimal value $\text{JR}({\mathcal{O}}_K)=4$, we deduce the undecidability of ${\mathcal{O}}_L$. This extends previous work which proved an analogous result in the opposite case $\text{JR}({\mathcal{O}}_K)=\mathrm{\infty }$. In particular, unlike prior work, we do not require that $L$ contains only finitely many roots of unity.