ON SIEGEL MODULAR FORMS OF GENUS TWO (II).1
ON SIEGEL MODULAR FORMS OF GENUS TWO (II).1
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DOI：
10.2307/2373172
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发表时间：
1964-04
影响因子：
1.7
通讯作者：
J. Igusa
中科院分区：
文献类型：
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作者：
J. Igusa
导言。我们将在第二篇文章中讨论的主要问题是具有层次的亏格2的Siegel模形式。我们在第一篇论文[5]中使用的方法即使对于第二级也不能提供足够的信息。因此，(由Grothendieck提出的)更高级别的模块化变体是否变得非单一的问题超出了我们的能力范围。考虑到其他一些应用，我们因此将“theta-常量”作为模形式进行了研究，并在最近的论文[6]中证明了其中的一个基本引理。利用文中的结果，我们将证明高水平模簇即使在其奇点附近也不存在非奇异覆盖。我们还将确定R2(1)/1‘2(2)=Sp(2，Z/2Z)如何作用于模型环A(R2(2))，并得到它对k=0，1，2，k的齐次部分A(R2(2))k的作用的特征。这样，我们将确定A(IF2(1))重现我们先前在A(R2(1))(2)上的结构定理。此外，还将得到四个基本的一阶Eisenstein级数的多项式与一个已知的此类恒等式(某一二阶Eisenstein级数与Riemann的Theta常数的8次方之间的恒等式)。我们注意到，这个恒等式以前是利用二次型上的Siegel主要定理得到的。
Introduction. The main subject we shall discuss in this second paper is the Siegel modular forms of genus two with levels. The method we used in the first paper [5] did not give sufficient information even for level two. Therefore the problem (raised by Grothendieck) whether modular varieties become non-singular or not for higher levels was beyond our reach. With some other applications in mind, we therefore investigated "theta-constants" as modular forms and proved, among others, a fundamental lemma in our recent paper [6]. Using the results in that paper, we shall show that modularvarieties of high levels do not have non-singular coverings even locally around their singular points. Also we shall determine how r2(1)/1'2(2) = Sp(2, Z/2Z) acts on the ring of modular forms A (r2 (2)) and obtain the characters of its action on the homogeneous parts A (r2 (2) ) k for k = 0, 1, 2, . In this way, we shall determine A (IF2 (1)) reproducing our earlier structure theorem on A (r2 (1 )) (2). Furthermore, the polynomial expressions of the four basic Eisenstein series of level one by theta-constants and a known identity of this kind (between a certain Eisenstein series of level two and the eighth power of Riemann's theta-constant [1]) will be obtained. We note that this identity was previously obtained using the Siegel main theorem on quadratic forms.