2 1. FROM QUADRATIC FORMS TO SPHERE PACKINGS AND COVERINGS

For example, the quadratic forms

(1.1)

x2

+

y2

and

x2

+ xy +

y2

are identified with

(1.2)

1 0

0 1

and

1

1

2

1

2

1

.

The space Sd is a

(

d+1

2

)

-dimensional real vector space. It becomes a Euclidean

space with the usual inner product

(1.3) Q, Q =

d

i,j=1

qij qij = trace(Q · Q ).

So in particular, we may identify

Sd

with

R(d+1).2

With the introduced terminology, many classical problems involving qua-

dratic forms fit into the following framework: Given a quadratic form Q ∈ Sd and

n ∈ N,

(1.4) does Q[x] = n have integral solutions x ∈

Zd

?

1.1.2. Arithmetical equivalence. When dealing with this problem in the

binary case (d = 2), Lagrange noticed that he could study the question up to

invertible, linear integral substitutions. Two quadratic forms Q, Q ∈

Sd

are called

arithmetically (or integral) equivalent, if there exists a U in the group

GLd(Z) = {U ∈

Zd×d

: | det U| = 1}

such that

Q = U

tQU.

Arithmetically equivalent forms provide the same set of values on integral vectors,

that is, Q [Zd] = Q[Zd], because Q [x] = Q[U x] and U Zd = Zd.

For example, the form

3x2

+ 3xy +

y2

is arithmetically equivalent to

x2

+ xy +

y2

because

(1.5)

−1 2

0 1

1

1

2

1

2

1

−1 0

2 1

=

3

3

2

3

2

1

.

Since for problem (1.4) it is suﬃcient to deal with one representative of each

equivalence class, it is desirable to know a unique form, or at least a small subset

of each class. These questions are addressed with in the theory of reduction (see

Chapter 2).

1.1.3. Positive definite quadratic forms. The theory of binary quadratic

forms, first systematically studied by Lagrange [162], splits into two branches,

depending on whether the forms are indefinite or definite. Because the roots of

the lattice sphere packing problem are to be found in the latter case, we restrict

ourselves to the definite case from now on.

A quadratic form, respectively real symmetric matrix Q ∈

Sd,

is positive defi-

nite, if Q[x] 0 for all x ∈

Rd

\{0}. The set of all positive definite quadratic forms

(PQFs from now on), respectively matrices, is denoted by

Sd

0

. Negative definite

forms are just the negative of positive forms and most of what follows just differs

by sign.