Kac-Moody Lie algebra and Hilbert modular forms

Kac-Moody 李代数和希尔伯特模形式

• 批准号: 14540022
• 负责人: TSUYUMINE Shigeaki
• 金额: $ 1.47万
• 依托单位: Mie University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540022/
• 关键词: theta series; totally real algebraic number field; quadratic forms; automorphic function; Weyl group; 総戻代数体; Hilbert; 保型形式; Hilbert保型形式; Fourier係数; Kac-Moody Lie環

Its is known that by integrating modular forms for SL_2(Z) with the kernel function which is theta series of indefinite quadratic form with polynomial, one obtain automorphic forms with singularities on Grassmann manifolds. There is a series of papers by R.E.Borcherds on this topic. Let A be the nonsingular integral symmetric matrix corresponding to the real part of variable of theta series, and let A_+ be the positive definite real symmetric matrix corresponding to the imaginary part. The matrix A_+ salifies the condition A_+A^<-1>A_+=A, and the whole of A_+ form the Grassmann manifold. Under some conditions of A, the singularities of the automorphic form determines Weyl chamber on the Grassmannians. It follows the relation between the automorphic forms and the Weyl groups of some Kac-Moody Lie algebras, and their denominator functions.Let K be a totally real algebraic number field, and let O_K be the ring of integers. In this research, we try to extend the all of the above argument to the case of Hilbert modular group SL_2(O_K). We show the inversion formula and transformation formulas for theta series with polynomial, of symmetric matrix A with coefficients in K. Further in the case that A has an anisotropic vector, we extend the theta series to the series involving theta series of quadratic forms of lower degree. This result is corresponding to Theorem 5.2 of Borcherds' paper "Automorphic forms with singularities oh Grassmanns", which is the key to obtaining automorphic form on Grassmanns. This result may be useful to study automorphic forms associated with Weyl chamber which is rational over K.

众所周知，通过将SL_2（Z）的模块化形式与内核函数集成在一起，该函数是Theta系列不确定的二次形式与多项式的形式，可以在Grassmann歧管上获得具有奇异性的自动形式。 R.E. Borcherds关于此主题有一系列论文。令A为与Theta系列变量的真实部分相对应的非介绍积分对称矩阵，让A_+为与假想部分相对应的正定确定的实际对称矩阵。矩阵A_+向条件A_+ A^<-1> A _+ = a征收，并且整个A_+形成Grassmann歧管。在A的某些条件下，自动形式的奇异性决定了拉士植物的韦尔室。它遵循自动形式与某些kac-moody lie代数的Weyl组之间的关系，其分母函数k是一个完全真实的代数数字字段，让O_K为整数。在这项研究中，我们尝试将上述所有论点扩展到希尔伯特模块化组SL_2（O_K）的情况。我们显示了具有多项式的Theta系列的反转公式和转换公式，对称矩阵A具有系数。该结果对应于Borcherds纸的定理5.2“具有奇异性oh Grassmanns”的纸质形式，这是在Grassmanns上获得自称形式的关键。该结果可能对研究与K型Weyl室相关的自态形式可能很有用。