In the 1960s Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct $$\chi _{5}$$χ5, the cusp form of lowest weight for the group $${\text {Sp}}(2,\mathbb {Z})$$Sp(2,Z). In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identified with the orthogonal group of signature (2, 3) for the lattice $$A_{1}$$A1 and Igusa’s form $$\chi _{5}$$χ5 appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three different types of constructions for modular forms. It turns out that our method produces simple coordinates.

中文翻译：

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$$A_{1}$$A1 塔的模块化形式

在 1960 年代 Igusa 确定了第二属的 Siegel 模形式的分级环。他使用 theta 级数构造了 $$\chi _{5}$$χ5，这是 $${\text {Sp}}(2,\mathbb {Z})$$Sp(2 ,Z)。2010 年 Gritsenko 发现了三座正交型模块化形式的塔，这些塔与某些系列的根格子相连。在这个设置中，Siegel 模形式可以用格 $$A_{1}$$A1 和 Igusa 的形式 $$\chi _{5}$$χ5 的签名 (2, 3) 的正交群来识别这座塔的。我们使用这种解释为这座塔构建了一个框架，该塔使用了三种不同类型的模块化结构。事实证明，我们的方法产生了简单的坐标。