Automorphic Forms and Lorentzian Kac-Moody Algebras. Part II
Automorphic Forms and Lorentzian Kac-Moody Algebras. Part II
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自守形式和洛伦兹 Kac-Moody 代数。
DOI：
10.1142/s0129167x98000117
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发表时间：
1996
期刊：
影响因子：
--
通讯作者：
V. Nikulin
中科院分区：
文献类型：
--
作者：
V. Gritsenko;V. Nikulin
我们给出了提升构造的变体，它在复数维度 3 的西格尔上半空间上定义了新的模形式，相对于完整的副模群（定义具有任意极化的阿贝尔曲面的模）。这些提升的数据是积分和半积分指数的雅可比形式。特别是，我们得到了模块化形式，它们是 Dedekind eta 函数的推广。其中一些形式定义了洛伦兹 Kac-Moody 代数与本文第一部分中分类的三阶双曲广义嘉当矩阵的自守校正。我们还构造了许多自同构形式，它们给出了 K3 表面的模量与皮卡德晶格条件的判别式。这些结果对于镜像对称和洛伦兹 Kac-Moody 代数理论很重要。 §0。简介 在第一部分中，我们发展了反射自同构形式的一般理论及其 IV 型埃尔米特对称域上的李反射自同构形式的特殊情况。这些自同构形式在镜像对称（对于 K3 和 Calabi-Yau ）和洛伦兹 Kac-Moody 代数中非常重要。在第一部分中，我们特别证明了该理论与双曲根系理论类似（它是其镜像对称变体）。我们解释过反射自同构形式是非常特殊的。据推测，它们的数量是有限的，类似于相应双曲根系的有限性结果，具有基本多面体（即椭圆形或抛物线型）的体积有限性的某些条件。我们相信并希望在进一步的出版物中表明，反射自同构形式和相应的双曲根系统的分类是卡拉比-丘的一些重要类别分类的关键步骤（参见[GN6]）。例如，椭圆形和抛物线型双曲根系以及反射自守形式的有限性结果与这些 Calabi-Yau 族的有限性相关。我们在第一部分中以对称（并扭曲为对称）双曲广义嘉当矩阵（具有 3 阶椭圆型且具有格子 Weyl 向量）为例，演示了双曲根系分类的一般方法。让我们用 A 来表示这些矩阵的集合。它包含 60 个矩阵。在第二部分中，我们考虑反射自同构的构造方法。特别是，对于许多广义嘉当 由京都大学 RIMS 资助 俄罗斯基础研究基金和京都大学 RIMS 资助
We give variants of lifting construction, which define new classes of modular forms on the Siegel upper half-space of complex dimension 3 with respect to the full paramodular groups (defining moduli of Abelian surfaces with arbitrary polarization). The data for these liftings are Jacobi forms of integral and half-integral indices. In particular, we get modular forms which are generalizations of the Dedekind eta-function. Some of these forms define automorphic corrections of Lorentzian Kac–Moody algebras with hyperbolic generalized Cartan matrices of rank three classified in Part I of this paper. We also construct many automorphic forms which give discriminants of moduli of K3 surfaces with conditions on Picard lattice. These results are important for Mirror Symmetry and theory of Lorentzian Kac–Moody algebras. §0. Introduction In Part I we developed the general theory of reflective automorphic forms and their particular case of Lie reflective automorphic forms on Hermitian symmetric domains of type IV. These automorphic forms are very important in Mirror Symmetry (for K3’s and Calabi–Yau’s) and for Lorentzian Kac–Moody algebras. In Part I, in particular, we showed that this theory is similar to the theory of hyperbolic root systems (it is its mirror symmetric variant). We explained that reflective automorphic forms are very exceptional. Conjecturally their number is finite similarly to finiteness results for corresponding hyperbolic root systems with some condition of finiteness of volume for fundamental polyhedron (i.e., of elliptic or parabolic type). We believe and hope to show in further publications that classification of reflective automorphic forms and corresponding hyperbolic root systems is the key step in classification of some important class of Calabi–Yau’s (see [GN6]). For example, finiteness results for hyperbolic root systems of elliptic and parabolic type and for reflective automorphic forms are related with finiteness of families of these Calabi–Yau’s. We demonstrated in Part I a general method of classification of the hyperbolic root systems on the example of symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type of rank 3 and with a lattice Weyl vector. Let us denote the set of these matrices by A. It contains 60 matrices. In this Part II we consider methods of construction of reflective automorphic forms. In particular, for many generalized Cartan Supported by RIMS of Kyoto University Supported by Grant of Russian Fund of Fundamental Research and RIMS of Kyoto University
影响因子：
1.7
作者：
J. Igusa
通讯作者：
J. Igusa