## On Siegel modular forms. II.(English)Zbl 0824.11031

[Part I, cf. J. Reine Angew. Math. 436, 57-85 (1993; Zbl 0772.11015).]
Author’s abstract: In this paper the author computes dimension formulas for rings of Siegel modular forms of genus $$g=3$$. Let $$\Gamma_ g (2)$$ denote the main congruence subgroup of level two, $$\Gamma_{g,0} (2)$$ the Hecke subgroup of level two and $$\Gamma_ g$$ the full modular group. The author gives the dimension formulas for genus $$g=3$$ for the above mentioned groups $$\Gamma$$ and determines the graded ring $$A(\Gamma_ 3 (2))$$ of modular forms with respect to $$\Gamma_ 3 (2)$$.
In part I the author computes the ring of modular forms for $$\Gamma_ 3 (2,4)$$ (the Igusa subgroup of level two). Principally this allows all rings of modular forms for subgroups $$\Gamma$$ to be computed with $$\Gamma_ 3 (2,4) \subset \Gamma \subset \Gamma_ 3$$. However, this involves subtle computations of rings of invariants with respect to the finite group $$\Gamma/ \Gamma_ 3 (2,4)$$. It turns out that the computation is simplified by constructing a certain central extension $$H_ g$$ of $$\Gamma_ g/ \Gamma_ g (2,4)$$. This group seems to be of independent interest because of its importance in coding theory. The main ingredient is a decomposition of Bruhat type for the group $$H_ g$$. This decomposition is closely connected with the theory of partial Fourier transformation.
Reviewer: B.Runge (Mannheim)

### MSC:

 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F27 Theta series; Weil representation; theta correspondences 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13A50 Actions of groups on commutative rings; invariant theory 11F03 Modular and automorphic functions

Zbl 0772.11015
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### References:

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