Kneser p-neighbors of Niemeier lattices,

By Gaetan Chenevier and Jean Lannes

Niemeier lattices.

The Niemeier lattices are the 24 isometry classes of even unimodular lattices in the euclidean space R24.
See for instance the Nebe & Sloane catalogue of lattices, as well as the references [B], [Ni] and [V] below.

# Root System Coxeter Number # Root System Coxeter Number # Root System Coxeter Number # Root System Coxeter Number
1 D24 46 7 A17E7 18 13 A92D6 10 19 D46 6
2 D16E8 30 8 A15D9 16 14 D64 10 20 A46 5
3 E83 30 9 D83 14 15 A83 9 21 A38 4
4 A24 25 10 A122 13 16 A72D52 8 22 A212 3
5 D122 22 11 A11D7E6 12 17 A64 7 23 A124 2
6 D10E72 18 12 E64 12 18 A54D4 6 24 Leech 0

Kneser Neighbors.

Let L and N be two even unimodular lattices in the Euclidean space R24, and let p>1 be a square-free integer.
Following M. Kneser, say that L and N are p-neighbors if their intersection has index p in L (hence in N).
An intruiguing number is then the number N(p,L,M) of p-neighbors of L which are isometric to M.
When p=2, all the N(2,L,M) have essentially been computed by R. Borcherds (see Nebe & Venkov [Ne-V]).

  • Following [C-L], here is the list of all [p, i, j, N(p,L(i),L(j))] where p is a prime less than or equal to 31 and L(k) is the k-th Niemeier lattice in the ordering above, that is #L(k)=k.

  • These computations, as well as all the ones below, were made with [Pari].

    P-neighborhood graph.

    The p-neighborhood graph of Niemeier lattices is the graph whose vertices are the 24 Niemeier lattices, labelled by their # = 1, 2, ..., 24 as above, with an edge between i and j if and only if L(i) has a p-neighbor isometric to L(j).
    As already said, when p is the prime 2 this graph was computed by R. Borcherds.
    In general, each of these graphs may be represented by a matrix of size 24 whose coefficient on raw i and column j is 1 if L(i) has a p-neighbor isometric to L(j), 0 otherwise.

  • Following [C-L], here is the list of all p-neighborhood graphs of Niemeier lattices, for p a squarefree integer.

  • 24 dimensional orthogonal Galois representations of conductor 1 and trivial coefficient, following [C-L].

  • The list of the 24 automorphic representations of level 1 and trivial coefficient for the orthogonal group over Z of any Niemeier lattice.

  • Some tables for the Hecke eigenvalues of the four Siegel eigenforms of genus 2 occuring in these formulas.

  • An explicit formula for N(p,L(i),L(j)).

  • References.

    [A] J. Arthur The endoscopic classification of representations: orthogonal and symplectic groups, preprint 2011.
    [B] R. Borcherds, The Leech lattice and other lattices, P.H.D. Thesis.
    [C-L] G. Chenevier & J. Lannes, Kneser neighbors and orthogonal Galois representations in dimensions 16 and 24, Oberwolfach Report, Algebraische Zahlentheorie (June 2011), a more complete version is coming soon.
    [Ne-V] G. Nebe & B. Venkov, On Siegel modular forms of weight 12, J. reine angew. Math. 351 (2001), p. 49--60.
    [Ni] H.-V. Niemeier, Definite quadratische Formen der Dimension 24 und Diskriminante 1, Journal of Number Theory 5 (1973).
    [Pari] The Pari-Group, Pari/GP.
    [V] B. Venkov's article in Conway & Sloane Sphere Packings, Lattices, and Groups, Springer-Verlag (1998).