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

利用我们的2桥结的彩色Jones多项式的公式Lawrence和Ron计算同调3球的SU(2)Witten—Reshetikhin—Turaev （WRT）不变量的方法，从结的彩色Jones多项式上得到了该不变量，我们推测了Ohtsuki不变量的一些数论性质。随后，我们利用habiro关于有色琼斯多项式的一些结果，给出了这个猜想。在此基础上，利用Hikami给出的Seifert 3-流形的Ohtsuki不变量的值，得到了积分同调3球的6次LMO不变量集，并对其进行了刻画。Lawrence和Zagier指出庞加莱同调球的WRT不变量与具有半积分权的向量模形式有关。即WRT不变量与模形式的Eichler积分的某一极限重合。作为这一结果的推广，我们计算了具有三种特殊纤维的球形Seifert流形的WRT不变量，并发现它与具有半积分权的向量模形式的Eicher积分的一个极限值相一致。利用模性质，得到了WRT不变量的精确渐近展开式，并从模形式的角度给出了Ohtsuki级数和Chern—Simons不变量等拓扑不变量的解释。此外，我们还指出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.