Siegel modular forms of weight 13 and the Leech lattice. (English) Zbl 1535.11070
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 \(n\geq 1\).
MSC:
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
| 11F30 | Fourier coefficients of automorphic forms |
| 11F27 | Theta series; Weil representation; theta correspondences |
| 20D08 | Simple groups: sporadic groups |
| 11H55 | Quadratic forms (reduction theory, extreme forms, etc.) |
Keywords:
Siegel modular forms; pluriharmonic theta series; Leech lattice; Golay code; Mathieu groupsReferences:
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