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.

通过将SL_2(Z)的模形式与核函数(不定二次型与多项式的级数)相结合，得到了Grassmann流形上具有奇异性的自同构形。关于这一主题，R.E.博尔切尔德发表了一系列论文。设A是theta级数中变量实部对应的非奇异积分对称矩阵，A+是虚部对应的正定实对称矩阵。矩阵A_+满足条件A_+A^&lt；-1&gt；A_+=A，且A_+的整体构成Grassmann流形。在A的某些条件下，自同构形的奇点决定了Grassmannians上的Weyl腔。研究了一些Kac-Moody李代数的自同构型与Weyl群及其分母函数之间的关系，设K为全实代数域，O_K为整数环.在本研究中，我们试图将上述所有论点推广到Hilbert模群SL_2(O_K)的情形。给出了系数为K的对称矩阵A的多项式θ级数的求逆公式和变换公式。进一步，在A具有各向异性向量的情况下，我们将其推广到含有低次二次型的θ级数。这一结果与Borcherds的“Grassmanns上具有奇点的自同构形”中的定理5.2相对应，该定理是获得Grassmanns上自同构形的关键。这一结果对于研究有理于K的Weyl腔的自同构形可能是有用的。