A result by Hashimoto and Ueda says that the graded ring of modular forms with respect to SO(2, 10) is a polynomial ring in modular forms of weights 4, 10,12,16,18, 22, 24, 28, 30, 36, 42. In this paper we show that one may choose Eisenstein series as generators. This is done by calculating sufficiently many Fourier coefficients of the restrictions to the Hermitian halfspace. Moreover we give two constructions of the skew symmetric modular form of weight 252.Let V be a real quadratic space of signature (2, 10). The bilinear form of V is denoted by (·, ·). The group of all isometries of V is called the orthogonal group of V and is given byBy SO(V ) we denote the subgroup of index two, which is equal to the kernel of the determinant-character. We obtain another subgroup O(V ) + of index two as the kernel of the real spinor norm. The intersection of the groups SO(V ) and O(V ) + is denoted by SO(V ) + . This is the connected component of the identity and is well-known to be a semisimple and noncompact Lie group, compare e.g. [15]. Its maximal compact subgroup is given by SO(2) × SO(10). In this text we fix L = E 8 to be the (up to isometries) unique even positive definite unimodular lattice in dimension 8. We denote the Gram matrix of L by S and define the even unimodular lattice of signature (2, 10)
In the 1960's Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct χ 5 , the cusp form of lowest weight for the group 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 and Igusa's form χ 5 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.
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