Rationality Problems for Chern - Simons Invariants

This paper makes certain observations regarding some conjectures of Milnor and Ramakrishnan in hyperbolic geometry and algebraic K-theory. As a consequence of our observations, we obtain new results and conjectures regarding the rationality and irrationality of Chern-Simons invariants of hyperbolic 3-manifolds. In this paper, by a hyperbolic 3-manifold, we shall mean a complete, oriented hyperbolic 3-manifold with nite volume. So a hyperbolic 3-manifold is compact or has nitely many cusps as its ends (see e.g. 20]). By a cusped manifold we shall mean a non-compact hyperbolic 3-manifold. A hyperbolic 3-manifold M is a quotient of the hyperbolic 3-space H 3 by a discrete subgroup ? of PSL 2 (C) with nite covolume. The isometry class of M determines the discrete subgroup up to conjugation. To each subgroup ? of PSL 2 (C), we can associate the trace eld of ?, that is, the subbeld of C generated by traces of all elements in ?. The trace eld clearly depends only on the conjugacy class of ?, so one can deene it to be the trace eld of the hyperbolic 3-manifold. However, the trace eld is not an invariant of commensurability class of ?, although it is not far removed. There is a notion of invariant trace eld due to Alan Reid (see 17]) which is a subbeld of trace eld and does give an invariant of commensurability classes. Let ? (2) be the subgroup of ? generated by squares of elements of ?. The invariant trace eld k(M) of M is deened to be the trace eld of ? (2). We will use invariant trace elds throughout this paper. We want to emphasize here that each invariant trace eld is a number eld together with a speciic embedding in C. Recall that a number eld is totally real if all its embeddings into C have image in R and totally imaginary if none of its embeddings has image in R. A CM-eld is a number eld with complex multiplication, i.e., it is a totally imaginary quadratic extension of a totally real number eld. A familiar class of CM-elds is the class of cyclotomic elds. We will also use a slightly more general notion. We say that an embedding : F , ! C of a number eld F is a CM-embedding if (F), as a subbeld of C , is an imaginary quadratic extension of a totally real eld. So …