On p-adic propreties of the Witten-Reshetikhin-Turaev invariant

We use the properties of the Melvin-Morton expansion of the colored Jones polynomial in order to prove that the trivial connection contribution converges p-adicly to the SO(3) Witten-Reshetikhin-Turaev invariant of rational homology spheres, as it was conjectured by R. Lawrence.