Distribution of Chern-Simons invariants

Let $M$ be a 3-manifold with a finite set $X(M)$ of conjugacy classes of representations $\rho :{\pi }_1(M)\to$SU${}_2$. We study here the distribution of the values of the Chern-Simons function CS$:X(M)\to \mathbb{R}\mathrm{/}2\pi \mathbb{Z}$. We observe in some examples that it resembles the distribution of quadratic residues. In particular for specific sequences of $3$-manifolds, the invariants tends to become equidistributed on the circle with white noise fluctuations of order $\mathrm{\mid }X(M){\mathrm{\mid }}^{-1\mathrm{/}2}$. We prove that for a manifold with toric boundary the Chern-Simons invariants of the Dehn fillings $M_{p\mathrm{/}q}$ have the same behaviour when $p$ and $q$ go to infinity and compute fluctuations at first order.