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arxiv: math-ph/0409016 · v1 · submitted 2004-09-08 · 🧮 math-ph · math.MP

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Quantum Invariant, Modular Form, and Lattice Points

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classification 🧮 math-ph math.MP
keywords eichlerinvariantintegralsmodularlatticenumberpointsasymptotic
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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 $N\to\infty$. We reveal that the number of dominating terms, which is the number of the non-vanishing Eichler integrals in a limit $\tau\to N\in\mathbb{Z}$, is related to that of lattice points inside 4-dimensional simplex, and we discuss a relationship with the irreducible representations of the fundamental group.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On Uniqueness of Mock Theta Functions

    math.NT 2026-04 unverdicted novelty 6.0

    Mock theta functions admit a unique resurgent continuation across their natural boundary, with the continuation fixed by their Mordell-Appell integrals via rotated Laplace contours.