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The cusp forms of weight $ 3$ on $ \Gamma\sb 2(2,4,8)$

Authors: Bert van Geemen and Duco van Straten
Journal: Math. Comp. 61 (1993), 849-872
MSC: Primary 11F55
MathSciNet review: 1181333
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Abstract: The congruence subgroup $ {\Gamma _2}(2,4,8)$ of the group $ {\Gamma _2}$ of $ 4 \times 4$ integral symplectic matrices is contained in $ {\Gamma _2}(4)$ and contains $ {\Gamma _2}(8)$, with $ {\Gamma _2}(n)$ the principal congruence subgroup of level n. The Satake compactification of the quotient of the three-dimensional Siegel upper half space by $ {\Gamma _2}(2,4,8)$ is shown to be a complete intersection of ten quadrics in $ {\mathbb{P}^{13}}$. We determine the space of global holomorphic three forms on this space, which coincides with the space of cusp forms of weight 3 on $ {\Gamma _2}(2,4,8)$; it has dimension 2283. Finally, we study the action of the Hecke operators on this space and consider the Andrianov L-functions of some eigenforms.

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Article copyright: © Copyright 1993 American Mathematical Society