Rings of automorphic forms which are not Cohen-Macaulay. II
Let \(\Gamma_n = \mathrm{Sp}_{2n}(\mathbb{Z})\) be the Siegel modular group of degree \(n\), and \(\Gamma_n(\ell)\) its congruence subgroup of level \(\ell\): \(\Gamma_n(\ell) = \{M\in \Gamma_n\mid M\equiv 1_{2n} \bmod \ell \}\). Let \(A(\Gamma_ n) = \oplus_{k\geq 0}A(\Gamma)_k\) be the graded ring of Siegel modular forms for \(\Gamma_n\), \(A(\Gamma)_k\) being the vector space of modular forms of weight \(k\). Let \(A(\Gamma)^{(r)}\) denote the ring \(\oplus_{k\equiv 0(r)} A(\Gamma)_k\) for an integer \(r\). In Part I [J. Math. Soc. Japan 38, 147--162 (1986; Zbl 0614.10026)] the present author obtained, together with related results, the result that \(A(\Gamma_n)^{(r)}\) is not Cohen-Macaulay (in abbreviation C.-M.) for any \(r\) if \(n\geq 4\). The result that \(A(\Gamma_1)^{(r)}\) and \(A(\Gamma_ 2)^{(r)}\) are C.-M. for any \(r\) is known. In the present paper the author treats the remaining case. The main result of this paper is that the ring \(A(\Gamma_3)^{(r)}\) is not C.-M. for any integer \(r\). The method of the proof is as follows. If \(A(\Gamma_3)^{(r)}\) is C.-M., then \(\dim H^6(X^*_3,\mathcal O_{X^*_3})\) must be equal to one, where \(X^*_3\) is the Satake compactification of the quotient space \(H_3/\Gamma_3\), \((H_3\) is the Siegel upper half space of degree 3). On the other hand, \(H^6(X^*_3, \mathcal O_{X_3^*})\) is shown to be \(\{0\}\) by making use of Igusa's desingularization. This is a contradiction.
scientific article; zbMATH DE number 4077335
scientific article