The Siegel Modular Group is the Lattice of Minimal Covolume in the Symplectic Group

Let $n⩾2$. We prove that, up to conjugation, ${\mathrm{S}\mathrm{p}}_{2n}(\mathbf{Z})$ is the unique lattice in ${\mathrm{S}\mathrm{p}}_{2n}(\mathbf{R})$ of the smallest covolume.