DISTRIBUTION OF CHERN-SIMONS INVARIANTS

Let M be a closed 3-manifold with a finite set X(M) of conjugacy classes of representations rho : pi(1)(M) -> SU2. We study here the distribution of the values of the Chern-Simons function CS : X (M) -> R/2 pi 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 vertical bar X(M)vertical bar(-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.