Research of relation among quantum invariant and number theoretic invariants and modular forms

应用我们给出的2桥纽结的有色Jones多项式的公式和Ron计算同调3-球面的SU(2)Witten-Reshetikhin-Turaev(WRT)不变量的方法，我们猜想了Ohtsuki不变量的一些理论性质.随后，利用K.Habiro给出的有色Jones多项式的一些结果，我们证明了猜想。此外，利用Hikami给出的Seifert 3-流形的Ohtsuki不变量的值，我们确定了积分同调3-球面的6次LMO不变量集，并刻画了6次以上的Ohtsuki不变量集。Lawrence和Zagier指出，Poincare同调球面的WRT不变量与具有半整数权的向量模形式有关。也就是说，WRT不变量与模形式的Eichler积分的某个极限重合。作为这一结果的推广，我们计算了具有三个例外纤维的球面Seifert流形的WRT不变量，发现它与具有半整数权的向量模形式的Eicher积分的一个极限值相吻合。利用模的性质，我们得到了WRT不变量的精确渐近展开式，并从模形式的角度解释了Ohtsuki级数和Chern-Simons不变量等拓扑不变量。此外，我们还指出了向量模形式与流形的基本群有关。

Applying to our formula of the colored Jones polynomial for 2-bridge knots Lawrence and Ron's way to compute SU(2)Witten--Reshetikhin--Turaev (WRT) invariant of homology 3-sphere obtained by surgery of a knot from the colored Jones polynomial of the knot, we conjectured some number theoretical properties of the Ohtsuki invariants. Later on, using some results about the colored Jones polynomial given by K.Habiro, we showed the conjecture. Furthermore, using the values of the Ohtsuki invariants for Seifert 3-manifolds given by Hikami, we identified the set of the LMO invariant up to degree 6 for integral homology 3-spheres, and characterized the set of the Ohtsuki invariants up to degree 6.Lawrence and Zagier pointed out that WRT invariant of the Poincare homology sphere is related to the vector modular form with half-integral weight. Namely the WRT invariant coincides with a certain limit of the Eichler integral of the modular forms. As a generalization of this result, we have computed the WRT invariant for the spherical Seifert manifold with three exceptional fibers, and we have found that it coincides with a limiting value of Eicher integral of vector modular form with half-integral weight. By use of the modular property, we have obtained an exact asymptotic expansion of the WRT invariant, and have given an interpretation of topological invariants such as the Ohtsuki series and the Chern--Simons invariant from the viewpoint of modular forms. Furthermore, we have pointed out that the vector modular form is related to the fundamental group of manifolds.