The ring of modular forms for the even unimodular lattice of signature (2,18)
A \({\mathbf U}\)-polarised \(K3\) surface (where \({\mathbf U}\) is the hyperbolic plane, the unique even unimodular lattice of rank 2) is a pair \((Y,j)\) where \(Y\) is a \(K3\) surface and \(j\colon {\mathbf U}\hookleftarrow \mathrm{Pic}Y\) is a primitive embedding (roughly, a elliptic \(K3\) with a section). Denote by \({\mathbf T}\) the even unimodular lattice \(2{\mathbf U}\oplus 2{\mathbf E}_8\) of signature \((2,18)\). It is well known that the coarse moduli space of \({\mathbf U}\)-polarised \(K3\) surfaces is a quotient of the symmetric domain associated with \({\mathbf T}\) by \(\Gamma=\mathrm{O}({\mathbf T})\). \par By studying this moduli space, the authors determine the ring of modular forms for \(\Gamma\). The ring \(A(\Gamma)\) of modular forms with trivial character is \({\mathbb C}[S]^{\mathrm{SL}_2}\) where \(S=\mathrm{Sym}^8\mathbb{C}^2\times \mathrm{Sym}^{12}\mathbb{C}^2\) with coordinates \(\mathbb{C}[S]=\mathbb{C}[u_{i,8-i},u_{j,12-j}]\) (with the indices being non-negative) and \(\mathrm{SL}_2\) acting with weight \((i+j)/2\) on \(u_{i,j}\). There are also modular forms of character \(\det\) and the total ring \(\tilde A[\Gamma]=\bigoplus_k\bigoplus_{\chi\in \{1,\det\}}A_k(\Gamma,\chi)\) is given by \[ \tilde A(\Gamma)\cong \mathbb{C}[S]^{\mathrm{SL}_2}[s_{132}]/(s_{132}^2-\Delta_{264}). \] Here \(\Delta_{264}\) is a form with trivial character that defines the reflection hyperplane of a \((-2)\)-vector, and \(s_{132}\) with character \(\det\) is given in [\textit{E. Freitag} and \textit{R. Salvati Manni}, J. Algebr. Geom. 16, No. 4, 753--791 (2007; Zbl 1128.11024)].
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