2 GABRIELE NEBE

had visiting professorships in Bonn (1989), Cambridge (1989), Geneva (1992, and 3

months every year since 1997), Lyon (1993), Paris (1994), Aachen (1994, 1996, ff.),

Berlin (1995/96), Grenoble (1995, 1998), Bordeaux (1997), Fukuoka (2002), Ky-

oto (2006). All groups enjoyed interesting and fruitful discussions and productive

collaborations with Boris Venkov. Some of the resulting articles are given in the

references; for a complete list of Boris Venkov’s papers I refer to the Zentralblatt

or MathSciNet.

2. Lattices, designs and modular forms

2.1. Extreme lattices. A classical problem asks for the densest packing of

equal spheres in Euclidean space. Already in dimension 3 this turned out to be

very hard; in this generality it was solved by T. Hales, 1998, with a computer

based proof of the Kepler conjecture. The sphere packing problem becomes easier

if one restricts to lattice sphere packings, where the centers of the spheres form a

group. The density function has only finitely many local maxima on the space of

similarity classes of n-dimensional lattices, the so called extreme lattices. Korkine

and Zolotareff and later Voronoi developed methods to compute all extreme lattices

of a given dimension. The necessary definitions are given in this section. Details

and proofs may be found in the textbook [31].

We always work in Euclidean n-space

(Rn,

(, )) where (x, y) =

∑n

i=1

xiyi is the

standard inner product with associated quadratic form

Q :

Rn

→ R,Q(x) :=

1

2

(x, x) =

1

2

n

i=1

xi

2.

Definition 2.1. (a) A lattice is the Z-span L = b1,...,bn

Z

=

{

∑n

i=1

aibi | ai ∈ Z} of a basis B = (b1,...,bn) of Rn.

(b) The determinant of L is the square of the covolume of L in Rn and can be

computed as the determinant of a Gram matrix det(L) = det((bi,bj))i,j=1.n

(c) L is called integral if (, m) ∈ Z for all , m ∈ L.

(d) L is called even if (, ) ∈ 2Z and hence Q( ) ∈ Z for all ∈ L.

(e) The minimum of L is min(L) = min{(x, x) | 0 = x ∈ L}. We denote the

set of minimal vectors by Min(L) := {x ∈ L | (x, x) = min(L)}.

(f) The sphere packing density of L is then proportional to the n/2-th power

of the Hermite invariant of L, γ(L) :=

min(L)

det(L)1/n

.

(g) A similarity of norm α ∈ R

0

is an element σ ∈ GLn(R) with (σ(x),σ(y)) =

α(x, y) for all x, y ∈

Rn.

Two lattices L and M are called similar if there

is a similarity σ with σ(L) = M.

(h) The Hermite function γ is well defined on similarity classes of lattices. A

lattice L is called extreme if its similarity class realises a local maximum

of γ.

The definition of extreme lattices goes back to Korkine and Zolotareff in the

1870s. They showed that extreme lattices are perfect, where a lattice L is per-

fect if the projections onto the minimal vectors span the space of all symmetric

endomorphisms, i.e.

xxtr

| x ∈ Min(L) = Rsym

n×n

.