Kneser neighbours and orthogonal Galois representations indimensions 16 and 24
Gaëtan Chenevier(joint work with Jean Lannes)
Let n ≥ 1 be an integer. Recall that an even unimodular lattice in the standard
euclidean space Rn is a lattice L ⊂ Rn of covolume 1 with x · x ∈ 2Z for all x ∈ L.
Let Xn denote the set of isometry classes of even unimodular lattices in Rn . As iswell-known, Xn is a finite set which is non-empty if and only if n ≡ 0 mod 8. For
example, the latticee1 + . . . + en,
En = Dn + Z2n{e1 , . . . , en } denoting the canonical basis
P of R and Dn the sublattice of index
n2 in Z whose elements (xi ) satisfyi xi ≡ 0 mod 2, is even unimodular forn ≡ 0 mod 8.
The set Xn has been determined in only three cases. One has X8 = {E8 }(Mordell), X16 = {E8 ⊕ E8 , E16 } (Witt) and Niemeier showed that X24 has 24
explicit elements (see [V]). The number of numerical coincidences related toNiemeier’s lists is quite extraordinary and makes that list still mysterious. Forthe other values of n the Minkowski-Siegel-Smith mass formula shows that Xn ishuge, perhaps impossible to describe. For instance, X32 already has more than
80.106 elements ([S]).
Let L ⊂ Rn be an even unimodular lattice, and let p be a prime; Kneser defines
a p-neighbour of L as an even unimodular lattice M ⊂ Rn such that M ∩ L has
index p in L (hence in M ). The relation of being p-neighbours turns Xn into agraph which was shown to be connected by Kneser, providing a theoretical way
to compute Xn from the single lattice En . This is actually the way Kneser andNiemeier computed Xn for n ≤ 24, using the prime p = 2 and the huge number ofsymmetries present in those cases.
In this paper, we are interested in giving an explicit formula for the number
Np (L, M ) of p-neighbours of L which are isometric to M . Equivalently, it amounts
to determining the Z-linear operator Tp : Z[Xn ] → Z[Xn ] defined by Tp [L] =P[N ], [−] denoting the isometry class of a lattice, the summation being over all
the p-neighbours of L.
Before stating our main results, let us mention that the p-neighbours of a given
even unimodular lattice L are in canonical bijection with the Fp -points of theprojective quadric CL over Z defined by the quadratic form x 7→ x·x
2 on L. The
quadric CL is hyperbolic over Fp for each prime p, thus L has exactlycn (p) = |CL (Fp )| = 1 + p + p2 + . . . + pn−2 + pn/2−1p-neighbours, where n = rkZ L. We are thus interested in the partition of thequadric CL (Fp ) into |Xn | parts (some of them being possibly empty) given by
the isometry classes. Of course, Np (E8 , E8 ) = c8 (p) as X8 = {E8 }, thus the firstinteresting case (perhaps known to specialists!) is n = 16.
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Theorem A : Let n = 16. In the basis E8 ⊕ E8 , E16 the matrix of Tp is11− τ (p) −405 286
1 02
3 1+pc16 (p)+ (1 + p + p + p ),0 1
405 −286
691PQnn 24
denoting Ramanujan’s ∆ function.
n≥1 τ (n)q = qn≥1 (1 − q )One first makes two well-known observations. First, the operators Tp commute
with each others. Second, they are self-adjoint for the scalar product defined byh[L], [M ]i = δ[L],[M ] |O(L)|, which amounts to saying that for all L, M ∈ Xn wehave(1)Np (L, M )|O(M )| = Np (M, L)|O(L)|.

Our main question (for any n) is thus equivalent to first finding a basis of R[Xn ]made of eigenvectors common to all of the Tp operators, and then to describing
the system of eigenvalues (λp ) of the (Tp ) on each of these eigenvectors. If n = 16
it is not difficult to compute T2 , and this was essentially done by Borcherds forn = 24 (see [N-V]). In both cases, the eigenvalues of T2 are distinct integers (thiswas noticed by Nebe and Venkov [N-V] for n = 24). Let us mention the importantpresenceof (cn (p)) as ”trivial” system of eigenvalues : formula (1) shows thatP−1[L]|O(L)|∈ Q[Xn ] is an eigenvector for Tp with eigenvalue cn (p).
L∈XnAssume now n = 16. The non-trivial system of eigenvalues is related to Ramanujan’s ∆-function in a non-trivial way. Our proof relies on Siegel theta seriesϑg : Z[Xn ] → M n2 (Sp2g (Z)),
the latter space being the space of classical Siegel modular forms of weight n/2 andgenus g. The generalized Eichler commutation relation ([R], [W]) asserts that ϑgintertwines Tp with some explicit Hecke operator on the space of Siegel modularforms. By a classical result of Witt, Kneser and Igusa (see [K]),ϑg (E8 ⊕ E8 ) = ϑg (E16 ) if g ≤ 3,
whereas F = ϑ4 (E8 ⊕ E8 ) − ϑ4 (E16 ) does not vanish, thus the Tp -eigenvalue weare looking for is related to the Hecke eigenvalues of F ∈ S8 (Sp8 (Z)). A result
by Poor and Yuen [P-Y] asserts that the latter space is 1-dimensional (generated
by the famous Schottky form!). But another non-trivial member of this spaceis Ikeda’s lift of Ramanujan’s ∆ function (see [I]), whose Hecke eigenvalues areexplicitly given in terms of ∆. By unravelling the precise formulae we obtain
Theorem A. Actually, we found a direct proof of the existence of Ikeda’s lift of ∆that relies on the triality for the reductive group PGO+
E8 over Z and two thetaseries constructions.
Pre-Theorem∗ B : There is an explicit formula as well for Tp if n = 24.
In this case it is more difficult to find the non trivial systems of eigenvalues on
Q[X24 ]. Five of them were actually identified as Ikeda lifts in the work of Nebeand Venkov [N-V], with a particular one due to Borcherds-Freitag-Weissauer. We
rather rely on Chapter 9 of the book [A] by Arthur . . . which is still unpublished at

Algebraische Zahlentheorie

3the moment; hence the ∗ in the statement above. The relation with automorphic
b whereforms comes from the canonical identification Xn = G(Q)\G(Af )/G(Z)G is the Z-orhogonal group of En , so that C[Xn ] is canonically the dual of thespace of automorphic forms of G of level 1 and trivial coefficients. The quickest
way (although perhaps inappropriate!) to state our results is in terms of Galois
representations.
Fix a prime `. Thanks to the works of many authors (including [A]), for any
system of eigenvalues π = (λp ) of (Tp ) on Q` [Xn ], there exists a unique continuous,semi-simple representationρπ,` : Gal(Q/Q) −→ GO(n, Q` )which is unramified outside ` and such that Trace(ρπ,` (Frobp )) = λp for each prime
p 6= `. This Galois representation is furthermore crystalline at ` with Hodge-Tate
numbers 0, 1, . . . , n − 2, the number n/2 − 1 occuring twice.
n−2 iFor example, if π = (cn (p)) then ρπ,` = (⊕i=0ω ) ⊕ ω n/2−1 , where ω is the `adic cyclotomic character. Moreover, Theorem A implies that if π is the non-trivial
system of eigenvalues when n = 16, then ρπ,` = ρ∆,` ⊗(⊕3i=0 ω i )⊕ω 7 ⊕(⊕6i=0 ω i )⊗ω 4 ,where ρ∆,` is Deligne’s Galois representation attached to ∆.
Consider now the following collection of orthogonal `-adic Galois representations of Gal(Q/Q) with similitude factor ω 22 and distinct Hodge-Tate numbers in{0, 2, . . . , 22}. For r odd and 1 ≤ r ≤ 23, set(2)i[r] = (⊕r−1i=0 ω ) ⊗ ω23−r
2.

For r even, k ∈ {12, 16, 18, 20, 22}, k + r ≤ 24, set(3)r−1 i∆k [r] = ρ∆k ,` ⊗ (⊕i=0ω )⊗ω

24−(k+r)2,where ∆k is a generator of Sk (SL2 (Z)). Set(4)Sym2 ∆ = Sym2 ρ∆,` .

For (j, k) ∈ {(6, 8), (4, 10), (8, 8), (12, 6)} then the space of vector-valued Siegel
modular cusp forms of genus 2 (for Sp4 (Z)) and coefficient Symj ⊗ detk has dimension one (Tsushima, see [VdG]). Let ∆j,k be a generator and let ρ∆j,k ,` be its
associated 4-dimensional `-adic Galois representation (Weissauer) ; it is symplectic
with Hodge-Tate numbers 0, k − 2, j + k − 1, and j + 2k − 3. Set(5)∆j,k [2] = ρ∆j,k ,` ⊗ (1 ⊕ ω) ⊗ ω

24−2k−j2.

Fact: There are exactly 24 representations of dimension 24 which are directsums of representations in the list (2)–(5) above and whose Hodge-Tate numbers
are 0, 1, . . . , 22 with 11 occuring twice.
For instance, Sym2 ∆ ⊕ ∆20 [2] ⊕ ∆16 [2] ⊕ ∆[2] ⊕ [9], ∆4,10 [2] ⊕ ∆18 [2] ⊕ ∆[4] ⊕[3] ⊕ [1], and Sym2 ∆ ⊕ ∆6,8 [2] ⊕ ∆16 [2] ⊕ ∆[2] ⊕ [5] are three of them. A more
precise form of pre-Theorem B (still conditionnal to Arthur’s results) is then thefollowing.

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Theorem∗ B : The Galois representations attached to the 24 systems of eigenvalues π occuring in Q[X24 ] are exactly the Galois representations above.
Thanks to this list, the only unknown to determine Tp on Z[X24 ] are the Heckeeigenvalues of the four Siegel cusp forms ∆j,k . They have actually been computed
by Van der Geer and Faber for all p ≤ 11, and even for some of them up to p = 37
: see [VdG]. We checked that their results fit with Borcherds’s computation of T2 .
Better yet, we directly proved that for any even unimodular root lattice L ⊂ R24with Coxeter number h(L) we have
Np (Leech, L) = 0 if p < h(L).
In particular, if L ∈ {E24 , E38 , E8 ⊕ E16 , A+24 } and p ≤ 23 then Np (Leech, L) = 0.
This simple fact allowed us to confirm all the eigenvalues given in the table of
Van der Geer and Faber for p ≤ 23 and to compute1 Tp for those p. Using theRamanujan estimates for the ∆k and ∆j,k , and with the convention h(Leech) = 1,
we obtain for instance :Corollary∗ : Assume p ≥ 11. If L and M are two even unimodular lattices
in R24 with Coxeter numbers h(L) ≥ h(M ), then Np (L, M ) ≥ 1 if, and only if,p ≥ h(L)/h(M ).
The author thanks Jean-Pierre Serre for his remarks.
References[A] J. Arthur, The endoscopic classification of representations: orthogonal and symplectic
groups, preprint available at http://www.claymath.org/cw/arthur/.
[V] B. Venkov, On the classification of integral even unimodular 24-dimensional quadraticforms, in J. H. Conway & N. J. Sloane, Sphere packings, lattices and groups, 3. ed.,Grundlehren der Math. Wissen. 290, Springer-Verlag, New York (1999).
[I] T. Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. ofMath. (2) 154 (2001), no. 3, 641681.
[K] M. Kneser, Lineare Relationen zwischen Darstellungsanzahlen quadratischer Formen, Math.
Ann. 168 (1967), 31–39.
[N-V] G. Nebe, B. & Venkov, On Siegel modular forms of weight 12, J. Reine Angew. Math.
531 (2001), 49–60.
[P-Y] C. Poor & D. S. Yuen, Dimensions of spaces of Siegel modular forms of low weight indegree four, Bull. Austral. Math. Soc. 54 (1996), no. 2, 309–315.
[R] S. Rallis, Langlands’ functoriality and the Weil representation, Amer. J. Math. 104 (1982),
no. 3, 469–515.
[S] J.-P. Serre, Cours d’arithmétique, P. U. F., Paris (1970).
[VdG] G. van der Geer, Siegel modular forms and their applications, The 1-2-3 of modular forms,
181–245, Universitext, Springer, Berlin (2008).
[W] L. Walling, Action of Hecke operators on Siegel theta series II, Int. J. Number Theory 4(2008), no. 6, 9811008.

1See http://www.math.polytechnique.fr/ chenevier/niemeier/niemeier.html for tables.

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