Kneser pneighbors 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
R^{24}.
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 
D_{24} 
46 
7 
A_{17}E_{7}

18 
13

A_{9}^{2}D_{6} 
10

19

D_{4}^{6} 
6

2 
D_{16}E_{8} 
30 
8 
A_{15}D_{9}

16 
14

D_{6}^{4} 
10

20

A_{4}^{6} 
5

3 
E_{8}^{3} 
30 
9 
D_{8}^{3} 
14 
15

A_{8}^{3}

9

21

A_{3}^{8} 
4

4 
A_{24} 
25 
10 
A_{12}^{2} 
13 
16

A_{7}^{2}D_{5}^{2} 
8

22

A_{2}^{12} 
3

5 
D_{12}^{2} 
22 
11 
A_{11}D_{7}E_{6}

12 
17

A_{6}^{4} 
7

23

A_{1}^{24} 
2

6 
D_{10}E_{7}^{2}

18 
12 
E_{6}^{4} 
12 
18

A_{5}^{4}D_{4} 
6

24

Leech

0

Kneser Neighbors.
Let L and N be two even unimodular lattices in the Euclidean space
R^{24}, and let p>1 be a squarefree
integer.
Following M. Kneser, say that L and N are pneighbors if their
intersection has index p in L (hence in N).
An intruiguing number is then the number
N(p,L,M) of pneighbors 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 [NeV]).
Following [CL], 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 kth Niemeier lattice
in the ordering above, that is #L(k)=k.
These computations, as well as all the ones below, were made with [Pari].
Pneighborhood graph.
The pneighborhood 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 pneighbor 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 pneighbor
isometric to L(j), 0 otherwise.
Following [CL], here is the list of all pneighborhood graphs of Niemeier
lattices, for p a squarefree integer.
24 dimensional orthogonal Galois representations of conductor 1 and trivial
coefficient, following [CL].
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.
[CL] 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.
[NeV] G. Nebe & B.
Venkov, On Siegel modular forms of weight
12, J. reine angew. Math. 351 (2001), p. 4960.
[Ni] H.V. Niemeier, Definite quadratische Formen der
Dimension 24 und Diskriminante 1, Journal of Number Theory 5
(1973).
[Pari] The PariGroup, Pari/GP.
[V] B. Venkov's article in Conway & Sloane Sphere Packings, Lattices, and
Groups, SpringerVerlag (1998).