Tau function approach to theta functions
Tau function approach to theta functions
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Tau 函数接近 theta 函数
DOI：
10.1093/imrn/rnv297
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发表时间：
2015
影响因子：
1
通讯作者：
Atsushi Nakayashiki
中科院分区：
文献类型：
--
作者：
徐俊庭;丸野健一;Bao-Feng Feng;太田泰広;Atsushi Nakayashiki
我们从孤子方程层次函数的角度研究了亏格黎曼曲面的theta函数。我们研究两种级数展开。一种是 θ 除数任意点的泰勒展开式。我们用 Schur 函数来描述展开的初始项，该函数对应于由某个扁线丛的间隙序列确定的分区。另一个是黎曼曲面上点的阿贝尔-雅可比图像上的变量之一的 theta 函数及其某些导数的展开。我们依次将展开式的初始项确定为theta函数的某些导数。作为副产品，我们首先获得了黎曼奇点定理的精化。其次，我们确定由 Korotkin 和 Shramchenko 定义的黎曼曲面的更高属西格玛函数的归一化常数，以便它们成为模不变量。
We study theta functions of a Riemann surface of genusfrom the view point of-function of a hierarchy of soliton equations. We study two kinds of series expansions. One is the Taylor expansion at any point of the theta divisor. We describe the initial term of the expansion by the Schur function corresponding to the partition determined by the gap sequence of a certain flat line bundle. The other is the expansion of the theta function and its certain derivatives in one of the variables on the Abel–Jacobi images ofpoints on a Riemann surface with. We determine the initial term of the expansion as certain derivatives of the theta function successively. As byproducts, firstly we obtain a refinement of Riemann's singularity theorem. Secondly we determine normalization constants of higher genus sigma functions of a Riemann surface, defined by Korotkin and Shramchenko, such that they become modular invariant.