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In this paper we engage in a general study of the asymptotic expansion of theWitten-Reshetikhin-Turaev invariants of mapping tori of surface mapping classgroup elements. We use the geometric construction of theWitten-Reshetikhin-Turaev TQFT via the geometric quantization of moduli spacesof flat connections on surfaces. We identify assumptions on the mapping classgroup elements that allow us to provide a full asymptotic expansion. Inparticular, we show that our results apply to all pseudo-Anosov mapping classeson a punctured torus and show by example that our assumptions on the mappingclass group elements are strictly weaker than hitherto successfully consideredassumptions in this context. The proof of our main theorem relies on our newresults regarding asymptotic expansions of oscillatory integrals, which allowsus to go significantly beyond the standard transversely cut out assumption onthe fixed point set. This makes use of Picard-Lefschetz theory for Laplaceintegrals.

Asymptotic expansions of the Witten-Reshetikhin-Turaev Invariants of Mapping Tori I