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On the quantum invariant for the Brieskorn homology spheres. (English) Zbl 1088.57013

The paper considers the asymptotic behaviour when \(N \rightarrow \infty\) of the Witten-Reshetikhin-Turaev quantum \(SU(2)\)-invariants \(\tau_N(M)\) for Brieskorn homology spheres \(M\). Starting (in section 2) from the explicit formula of \(\tau_N(M)\) computed by R. J. Lawrence and L. Rozansky in [Commun. Math. Phys. 205, 287–314 (1999; Zbl 0966.57017)], the invariants are shown to coincide, up to a suitable rescaling, with a limiting value of the Eichler integral of a half-integer weight modular form (section 5, theorem 9). The result follows from arguments of R. J. Lawrence and D. B. Zagier in [Asian J. Math. 3, No. 1, 93–107 (1999; Zbl 1024.11028)], in the case of the Poincaré homology sphere. Using a nearly modular property of the Eichler integral, the leading term when \(N \rightarrow \infty\) of the asymptotic expansion of \(\tau_N(M)\) is derived, and shown to be in agreement with the stationnary phase approximation of the WRT invariants. The paper concludes with examples, and discusses relations with Ohtsuki’s series. This work is further developed by the author in [Int. Math. Res. Not. 2005, No. 3, 121–154 (2005; Zbl 1075.57005)].

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
58J37 Perturbations of PDEs on manifolds; asymptotics
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