Quantum invariant, modular form, and lattice points

We study the Witten-Reshetikhin-Turaev SU(2) invariant for the Seifert manifold with 4-singular fibers. We define the Eichler integrals of the modular forms with half-integral weight, and we show that the invariant is rewritten as a sum of the Eichler integrals. Using a nearly modular property of the Eichler integral, we give an exact asymptotic expansion of the WRT invariant in N → ∞. We reveal that the number of dominating terms, which is the number of the nonvanishing Eichler integrals in a limit τ→ N ∈ℤ, is related to that of lattice points inside 4-dimensional simplex, and we discuss a relationship with the irreducible representations of the fundamental group.