New construction of vector bundles on Riemann surfaces and Verlinde's formula

1.证明了Riemann曲面上靠近极大退化曲线的0次稳定向量丛是由使Riemann曲面一致的肖特基群的线性表示得到的。利用Abel-Jacobi定理和Verlinde公式刻画了肖特基群的表示空间与向量丛的模空间之间的关系。我们刻画了包含1/6的环上的2次Siegel模形式的环结构，推广了Igusa的结果。进一步，我们将Swinnerton-Dyer，Serre和Katz关于椭圆模形式的同余和p-进性质的结果推广到Siegel模形式的情形。给出了抽象Wiener空间中微扰Chern-Simons积分单圈近似的严格数学模型，并利用抽象Wiener空间上的Malliavin-Taniguchi换变量公式，导出了Chern-Simons积分关于电荷参数的渐近展开式。

1. We showed that any stable vector bundle of degree 0 on a Riemann surface close to a maximally degenerate curve is obtained from a linear representation of the Schottky group uniformizing the Riemann surface. Further, we described the relationship between the representation space of the Schottky group and the moduli space of the vector bundles by using Abel-Jacobi's theorem and Verlinde's formula.2. We described the ring structure of Siegel modular forms of degree 2 over a ring containing 1/6 extending Igusa's result. Further, we extended results of Swinnerton-Dyer, Serre and Katz on congruence and p-adic properties of elliptic modular forms to the case of Siegel modular forms.3. We gave a mathematical rigorous model of the one loop approximation of the perturbative Chern-Simons integral in an abstract Wiener space setting, and by appealing to the Malliavin-Taniguchi formula of changing variables on the abstract Wiener space, we derived the asymptotic expansion for the Chern-Simons integral with respect to the charge parameter.