On Hilbert modular forms. III (Q760459)

Let \(A_{{\mathbb{Z}}}(\Gamma_ K)_ k\) be the \({\mathbb{Z}}\)-module of symmetric Hilbert modular forms of (integral) weight k with Fourier coefficients in \({\mathbb{Z}}\), \(A_{{\mathbb{Z}}}(\Gamma_ K):=\oplus_{r\geq 0}A_{{\mathbb{Z}}}(\Gamma_ K)_{2r}\) and \(A^ a_{{\mathbb{Z}}}(\Gamma_ K):=\oplus_{k\geq 0}A_{{\mathbb{Z}}}(\Gamma_ K)_ k\). In part II [ibid. 58, 44-46 (1982; Zbl 0507.10021)], the author has described the struture of the graded ring \(A_{{\mathbb{Z}}}(\Gamma_ K)\) for \(K={\mathbb{Q}}(\sqrt{2})\), \({\mathbb{Q}}(\sqrt{5})\), in terms of modular forms explicitly given by Eisenstein series. This paper, concerned first with \(A^ a_{{\mathbb{Z}}}(\Gamma_ K)\) for \(K={\mathbb{Q}}(\sqrt{2})\), adapts Resnikoff's method in the case \(K={\mathbb{Q}}(\sqrt{5})\) (based on Igusa-Hammond's modular imbedding and) giving the existence of a symmetric Hilbert modular form of weight 15. The author presents a minimal set of 4 generators for \(A^ a_{{\mathbb{Z}}}(\Gamma_ K)\) for \(K={\mathbb{Q}}(\sqrt{2})\); he also describes a minimal set of 5 generators (including the W above) for the case \(K={\mathbb{Q}}(\sqrt{5})\). The results are consistent with earlier work of Hirzebruch on the ring of Hilbert modular forms and also of H. Cohn, for \(K={\mathbb{Q}}(\sqrt{2})\).