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

Runge, Bernhard

Nagoya Math. J. 138, 179-197 (1995).

[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

Keywords:

invariant theory; Poincaré series; Gorenstein; dimension formulas; Siegel modular forms; decomposition of Bruhat type; partial Fourier transformation

Citations:

Zbl 0772.11015

References:

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