The graded ring of modular forms on the Cayley half-space of degree two

Abstract

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)

The classification of free algebras of orthogonal modular forms

“…The algebra of modular forms for every Γ R above is free and the constructions of generators are known. It was proved in [HU14] that the algebra of modular forms on O + (2U ⊕ E 8 (−1)) is freely generated by forms of weights 4, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42, and in [DKW19] that the generators can be chosen as additive lifts of Jacobi Eisenstein series. The other 25 cases were proved in a universal and elementary way in [WW20].…”

On some free algebras of orthogonal modular forms

“…However, it was announced in [HU14] that the graded ring of modular forms on O + (II 2,2 ⊕E 8 (−1)) is freely generated by forms of weights 4, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42. Assuming the structure result of [HU14], it was shown in [DKW19] that the generators can be constructed as the additive lifts of Jacobi Eisenstein series.…”

On some free algebras of orthogonal modular forms

“…However, it was announced in [HU14] that the graded ring of modular forms on O + (II 2,2 ⊕E 8 (−1)) is freely generated by forms of weights 4, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42. Assuming the structure result of [HU14], it was shown in [DKW19] that the generators can be constructed as the additive lifts of Jacobi Eisenstein series.…”