Distribution of Chern–Simons invariants

Let M be a 3-manifold with a finite set X(M) of conjugacy classes of representations ρ : π1(M) → SU2. We study here the distribution of the values of the Chern-Simons function CS : X(M) → R/2πZ. We observe in some examples that it resembles the distribution of qua-dratic residues. In particular for specific sequences of 3-manifolds, the invariants tends to become equidistributed on the circle with white noise fluctuations of order |X(M)| −1/2. We prove that for a manifold with toric boundary the Chern-Simons invariants of the Dehn fillings M p/q have the same behaviour when p and q go to infinity and compute fluctuations at first order.