The ring of modular forms of \(\mathrm{O}(2,4;\mathbb{Z})\) with characters
Consider the lattice \(P_1 := U\oplus E_7^{\oplus 2}\), which is a primitive sublattice of the \(K3\) lattice \(\Lambda_{K3}\). Let \(\mathcal{D}\) be the bounded Hermitian domain of type \(IV\) associated to \(T_1\). The orthogonal complement of \(P_1\) in \(\Lambda_{K3}\) is given by \(T_1 = U^{\oplus 2}\oplus A_1^{\oplus 2}\), which is naturally embeded with index \(2\) into the odd lattice \(T_2 = U^{\oplus 2}\oplus \langle -1\rangle^{\oplus 2}\). In fact, we have \(O(T_2) \simeq O(2,4;\mathbb{Z})\). The aim of the paper under review is to describe the graded ring \(\tilde{A}(O(2,4;\mathbb{Z}))\) of modular forms on \(\mathcal{D}\) with all characters of the group \(O(2,4;\mathbb{Z})\) in terms of a polynomial ring of 8 weighted variables with three relations. In this case, the Cox ring of the quotient stack \(\mathbb{M}\) of the coarse moduli space \(M := \mathcal{D}\slash O(T_1)\) of \(P_1\)-polarized \(K3\) surfaces is isomorphic to \(\tilde{A}(O(2,4;\mathbb{Z}))\). So, it is necessary to describe the Cox ring of the quotient stack \(\mathbb{M}\). In the proof of the main theorem, the authors use the projective model of a \(P_1\)-polarized \(K3\) surface as a Jacobian elliptic surface with singular fibres of type \(II^*\) and \(III^*\).
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