Siegel modular forms of weight 13 and the Leech lattice

Gaëtan Chenevier (1, 2) , Olivier Taïbi (3)

1 LMO - Laboratoire de Mathématiques d'Orsay
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

For g = 8, 12, 16 and 24, there is an alternating g-multilinear form on the Leech lattice, unique up to 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 Sp 2g (Z). We prove that they are nonzero eigenforms, determine one of their Fourier coefficients, and give informations about their standard 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 Sp 2n (Z), for any n ≥ 1.

Domaines: Mathématiques [math]; Théorie des nombres [math.NT]; Théorie des groupes [math.GR]; Géométrie algébrique [math.AG]

Gaëtan Chenevier, Olivier Taïbi. Siegel modular forms of weight 13 and the Leech lattice. L'Enseignement Mathématique , In press. ⟨hal-02265676⟩