Siegel modular forms of weight 13 and the Leech lattice

Abstract

For $g=8,12,16$ and $24$, there is a nonzero alternating $g$-multilinear form on the $\mathrm{Leech}$ lattice, unique up to a scalar, which is invariant by the orthogonal group of $\mathrm{Leech}$. The harmonic Siegel theta series built from these alternating forms are Siegel modular cuspforms of weight $13$ for ${\mathrm{Sp}}_{2g}(\mathbb{Z})$. We prove that they are nonzero eigenforms, determine one of their Fourier coefficients, and give informations about their standard $\mathrm{L}$-functions. These forms are interesting since, by a recent work of the authors, they are the only nonzero Siegel modular forms of weight $13$ for ${\mathrm{Sp}}_{2n}(\mathbb{Z})$, for any $n\ge 1$.