Siegel modular forms of weight 13 and the Leech lattice (Q2154803)

Summary: For \(g=8,12,16\) and \(24\), there is a nonzero alternating \(g\)-multilinear form on the Leech lattice, unique up to a scalar, which is invariant by the orthogonal group of 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 \(n\geq 1\).