On \(Q_{ab}\)-rationality of Eisenstein series of weight 3/2 (Q1262886)

Let F be a totally real algebraic number field. Shimura proved in a general framework that the orthogonal complements of the spaces of cusp forms in the spaces of holomorphic Hilbert modular forms of integral or half-integral weight over F are generated by \(Q_{ab}\)-rational ones except in the following two cases: (1) \(F={\mathbb{Q}}\) and the weight is 2; (2) \(F={\mathbb{Q}}\) and the weight is 3/2. [\textit{G. Shimura} Math. Ann. 260, 269-302 (1982; Zbl 0502.10013)]. In the first exceptional case, the Fourier coefficients of Eisenstein series are classically well known and the assertion is true. In the paper under review the author proves the above assertion for the second exceptional case. The reviewer has constructed bases of the spaces of Eisenstein series of weight 3/2 [Trans. Am. Math. Soc. 274, 573-606 (1982; Zbl 0503.10016); ibid. 283, 589-603 (1984; Zbl 0535.10026)]. One can also verify the above assertion using the bases and the Proposition 1.5 of \textit{G. Shimura} [Duke Math. J. 43, 673-696 (1976; Zbl 0371.14022)]. But the author's proof based on the results of Shimura [Rev. Mat. Iberoam. 1, No.3, 1-42 (1985; Zbl 0608.10028)] is more conceptual and shorter.