Clozel–Buzzard–Gee conjecture on C-algebraicity and fields of rationality

Let $G$ be a connected reductive group over $\mathbb{Q}$ and let $\pi$ be an automorphic representation of $G(\mathbb{A})$. The field of rationality $\mathbb{Q}(\pi)$ is the field over which the isomorphism class of $\pi^\infty$ is defined. Clozel–Buzzard–Gee conjecture. $\pi$ is C-algebraic if and only if $\mathbb{Q}(\pi)$ is finite over $\mathbb{Q}$. This predicts that C-algebraicity is exactly the condition ensuring a number field of rationality; the paper proves finiteness for cohomological representations, while the general equivalence remains open.