The octavic modular group

Let \( \mathrm{II}_{2,10}\) be the even unimodular lattice of rank 12 and signature (2,10). Its spin group is called the octavic modular group. In this talk, we will give an explicit embedding of the octavic modular group into the product of two copies of the Siegel modular group \(\Gamma_{16} = \mathrm{Sp}(32,\mathbb Z)\). We also present an embedding of the corresponding half-spaces. It turns out that the image of the octavic modular group is contained in the theta group \(\Gamma_{\theta,16} =\Gamma_{16} (1,2)\), so the construction can be used to obtain orthogonal modular forms from theta series.