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Graded rings of Hermitian modular forms of degree 2. (English) Zbl 1038.11030

Let \(K\) be an imaginary quadratic field and \({\mathcal O}_K\) its ring of integers. The Hermitian modular group \(\Gamma_2({\mathcal O}_K)\) of degree 2 over \(K\) consists of all \(4\times 4\) matrices \(M\) with entries in \({\mathcal O}_K\) which satisfy \(MJ\overline{M}^{\text{tr}}= J\) where \(J= \left(\begin{smallmatrix} 0 & -I\\ I & 0\end{smallmatrix}\right)\) and \(I\) is the \(2\times 2\) identity matrix. It acts on the Hermitian half space \(H_2(\mathbb{C})\) which consists of all \(2\times 2\) complex matrices \(Z\) for which \({1\over 2i} (Z- \overline Z^{\text{tr}})\) is positive definite. The submanifold of symmetric matrices in \(H_2(\mathbb{C})\) is the Siegel half-space \(H_2(\mathbb{R})\). The Siegel modular group \(\text{Sp}_2(\mathbb{Z})\) is a subgroup in \(\Gamma_2({\mathcal O}_K)\) for every K. J. Igusa [Am. J. Math. 86, 219–246 (1964; Zbl 0146.31703)] first described the graded ring of Siegel modular forms of degree 2 as a polynomial ring with explicitly specified generators. Then in 1967 E. Freitag proved an analogous result for \(\Gamma_2({\mathcal O}_K)\) when \(K= \mathbb{Q}(\sqrt{-4})\) is the Gaussian number field. Partial results for \(K= \mathbb{Q}(\sqrt{-3})\) were obtained by E. Freitag and C. F. Hermann [Adv. Math. 152, 203–287 (2000; Zbl 0974.11028)].
In the present paper, the authors give complete descriptions of the graded rings of modular forms on \(\Gamma_2({\mathcal O}_K)\) for \(K= \mathbb{Q}(\sqrt{-4})\) and \(K= \mathbb{Q}(\sqrt{-3})\) with respect to all Abelian characters of these groups. Their first result states that any Siegel modular form of degree 2 and even weight can be lifted to a Hermitian modular form of degree 2 over any imaginary quadratic number field \(K\). Then some new methods are developed to construct Hermitian modular forms of the appropriate kind. Maass lifts are used to construct skew-symmetric Hermitian modular forms, and Borcherds products provide Hermitian modular forms whose divisor sets are known. Then a reduction process as used by Igusa and Freitag yields the structure theorems. [For Borcherds products, see R. E. Borcherds, Invent. Math. 132, 491–562 (1998; Zbl 0919.11036) and V. A. Gritsenko and V. V. Nikulin, Sb. Math. 187, 1601–1641 (1996; Zbl 0876.17026)]. The structure theorems are stated in terms of generators and relations. Theta constants can be chosen as generators. As an application, the structure of the fields of Hermitian modular functions of degree 2 for \(K= \mathbb{Q}(\sqrt{-4})\) and \(K= \mathbb{Q}(\sqrt{-3})\) is determined. The subfields of symmetric Hermitian modular functions are rational function fields with four generators which are explicitly given in terms of Eisenstein series and Borcherds products.

MSC:

11F55 Other groups and their modular and automorphic forms (several variables)
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F27 Theta series; Weil representation; theta correspondences
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