6 1. FROM QUADRATIC FORMS TO SPHERE PACKINGS AND COVERINGS

Figure 4. Two bases of the hexagonal lattice obtained from the

PQFs in Figure 1.

The volume is called the determinant of the lattice L =

AZd

and denoted by det L.

This is justified, since it does not depend on the choice of the particular basis. For

the PQF Q =

AtA

we have det L =

√

det Q.

1.2.2. Hermite’s constant. The lattice interpretation of Gauss opened the

door for the use of geometric arguments in the theory of quadratic forms. A classical

example is the following theorem and its proof.

Theorem 1.1 (Hermite, [135]). Every lattice L has a base A = (a1, . . . , ad) ∈

GLd(R), such that

a1···ad ≤

4

3

d(d−1)/4

det L.

Using the dictionary, the theorem gives an upper bound for the arithmetical

minimum, depending on the determinant of a given PQF Q:

Corollary 1.2. λ(Q) ≤ (det

Q)1/d

·

(

4

3

)(d−1)/2

for Q ∈

Sd

0

.

The corollary in particular implies the existence of Hermite’s constant

(1.12) Hd = sup

Q∈Sd

0

λ(Q)

(det Q)1/d

.

Hermite’s constant and generalizations have been extensively studied, e.g. in the

context of algebraic number theory and differential geometry. We refer to [32],

[220], [70] and [265] for further reading.

1.2.3. Lattice sphere packings. Hermite’s theorem 1.1 and its proof (cf.

[174]) show how helpful the geometric viewpoint can be to obtain arithmetic results

as Corollary 1.2. The following sphere packing interpretation due to Minkowski

exemplifies this even more.

For a PQF Q =

AtA

and a lattice L =

AZd,

λ(L) =

λ(Q)

2

is called the packing radius of L. Thinking of solid spheres of radius λ(L) around

each lattice point, the spheres do not overlap and form a (lattice) sphere packing

(1.13) L +

λ(L)Bd

= {v + λ(L)x : v ∈ L, x ∈

Bd}.

See Figure 5 for portions of two famous examples. Due to this interpretation of the

arithmetical minimum, the number | Min Q| of its representatives is also known as