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 ${\rm Leech}$ lattice, unique up to a scalar, which is invariant by the orthogonal group of ${\rm Leech}$. The harmonic Siegel theta series built from these alternating forms are Siegel modular cuspforms of weight $13$ for ${\rm Sp}_{2g}(\mathbb{Z})$. We prove that they are nonzero eigenforms, determine one of their Fourier coefficients, and give informations about their standard ${\rm L}$-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 ${\rm Sp}_{2n}(\mathbb{Z})$, for any $n\geq 1$.