Generalized Gaussian sums Chern-Simons-Witten-Jones invariants of len-spaces

Starting from evaluating all Gaussian sums, we calculate (tau(r), tau greater than or equal to 2) (in the 4r-th cyclotomic fields) and {xi(r), r odd greater than or equal to 3} (in the r-th cyclotomic fields) for all lens spaces L(p,q). We prove that they are all algebraic integers and show that xi(r) determines the Dedekind sum s(q,p), and hence determines the generalized Gasson invariant of lens space. We conjecture these two properties hold for more general 3-manifold, and some evidences are discussed. Though the formulae are not simple, we are able to give necessary and sufficient conditions for lens spaces to have the same CSWJ invariants. Examples of lens spaces with the same invariants but different topological types are given. Some applications in number theory are also included.