On the Quantum Invariant for the Brieskorn Homology Spheres

We study an exact asymptotic behavior of the Witten–Reshetikhin–Turaev invariant for the Brieskorn homology spheres Σ(p1, p2, p3) by use of properties of the modular form following a method proposed by R. Lawrence and D. Zagier. Key observation is that the invariant coincides with a limiting value of the Eichler integral of the modular form with weight 3/2. We show that the Casson invariant is related to the number of the Eichler integrals which do not vanish in a limit τ → N ∈ Z. Correspondingly there is a one-to-one correspondence between the non-vanishing Eichler integrals and the irreducible representation of the fundamental group, and the Chern–Simons invariant is given from the Eichler integral in this limit. It is also shown that the Ohtsuki invariant follows from a nearly modular property of the Eichler integral, and we give an explicit form in terms of the L-function.