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
(, )) where (x, y) =
xiyi is the
standard inner product with associated quadratic form
Q :
R,Q(x) :=
(x, x) =
Definition 2.1. (a) A lattice is the Z-span L = b1,...,bn
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) :=
(g) A similarity of norm α R
is an element σ GLn(R) with (σ(x),σ(y)) =
α(x, y) for all x, y
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.
| x Min(L) = Rsym
Previous Page Next Page