Abstract
First we deal with the notion of d-neighbors (d positive integer, very often prime) for even unimodular lattices, introduced by M. Kneser, and with the associated Hecke operators; numerous examples are given. Then we analyse in depth the d-neighborhoods between a Niemeier lattice with roots and the Leech lattice; this sheds some light on the “holy constructions” of the latter by Conway and Sloane. At the very end of the chapter we describe an iterative 2-neighbor algorithm, essentially due to Borcherds, which starting from any even unimodular lattice of dimension 24 produces the Leech lattice after at most 5 steps.
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Chenevier, G., Lannes, J. (2019). Kneser Neighbors. In: Automorphic Forms and Even Unimodular Lattices. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-319-95891-0_3
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DOI: https://doi.org/10.1007/978-3-319-95891-0_3
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