The irrationality conjecture for Chern–Simons invariants

Let $M$ be a hyperbolic $3$-manifold with invariant trace field $k=k(M)$, and let $\overline{k}$ denote its complex conjugate. The Chern–Simons invariant $CS(M)$ lies in $\mathbb{R}/\pi^2\mathbb{Z}$ and is called irrational when it does not lie in $\pi^2\mathbb{Q}/\pi^2\mathbb{Z}$. Irrationality conjecture. If

$$k\cap\overline{k}\subset\mathbb{R},$$

then $CS(M)$ is irrational. In particular, $CS(M)$ is irrational if $k(M)$ has odd degree over $\mathbb{Q}$. Numerical evidence suggests that Chern–Simons invariants are usually irrational outside the CM-embedding case, although irrationality has not been proved for any example; the conjecture remains open.