We now jump from speciﬁc examples of Galois repre-

sentations to an important general m ethod for ﬁnding

them. This method, using “

´

etale cohomology,” sets up a

direct relationship between systems of Z-equations and

representations of the absolute Galois group G.Thisre-

lationship comes from some very advanced mathematical

constructions whose nature we can describe only in a

very sketchy way. The discovery (or invention) of

´

etale

cohomology is one of the major successes of twentieth-

century mathematics.

Vector Spaces and Linear Actions of Groups

We have seen in preceding chapters examples of one- and two-

dimensional Galois representations. But it is very unlikely that

they include all the information to be found in the absolute Galois

group of Q. Indeed they do not. We seek a large supply of Galois

representations of all dimensions.

We need a general method to derive linear Galois representations

from Z-varieties. There are several of these methods, but we discuss

just one: ´etale cohomology. It is too advanced to explain in detail

226 CHAPTER 20

here, but we can give a very rough idea of what is going on. First, a

deﬁnition:

DEFINITION: If k is a ﬁeld and n ≥ 1, then an n-dimensional

vector over k is an n-by-1 matrix with entries from k.Theset

of all n-dimensional vectors over k is denoted k

n

, and is called

a vector space.

For example, C

3

contains the three-dimensional vector

⎡

⎢

⎣

3

i

√

2

⎤

⎥

⎦

.

The vector

⎡

⎢

⎢

⎢

⎣

2

4

3

−10

⎤

⎥

⎥

⎥

⎦

is contained in Q

4

, and is also in R

4

.

DEFINITION: A line in k

n

is a subset of k

n

consisting of all

vectors that are proportional to some given vector.

An example of a line in C

3

is the set of all the three-dimensional

vectors

⎡

⎢

⎣

3c

ic

√

2c

⎤

⎥

⎦

,

where c takes on all possible values in C (including 0—the 0-vector

is in every line by this deﬁnition.)

A linear representation, you will remember, is a morphism from

agroupG to a group of square matrices GL(n, k), where k is some

number system. We call this a “linear” representation because if M

is a matrix and L is a line in the n-dimensional space of vectors k

n

,

then ML is always a line again.

A MACHINE FOR MAKING GALOIS REPRESENTATIONS 227

For example, if M is the matrix

12

45

in GL(2, Q)andL is the

line of all vectors

a

b

such that b = 2a (if you graphed this, it would

be the line of slope 2 through the origin) then ML is the line of all

vectors

c

d

,whered = (14/5)c.

Why? When we multiply the matrix M by the vector

a

b

we get

a + 2b

4a + 5b

.Nowif

a

b

were a point on the line L to begin with, that

would mean that b = 2a. If we substitute this in, we get

a + 2b

4a + 5b

=

a + 4a

4a + 10a

=

5a

14a

. This last vector is of the form

c

d

where d =

14

5

c.

Thus M sends the line of slope 2 to the line of slope

14

5

. Simi-

larly, you can check that M sends any line through the origin to

another line through the origin. And this is why we call v → Mv a

linear map.

Now suppose M is any n-by-n matrix in GL(n, k)andv and w are

any vectors in k

n

,anda is any number in k.ThenM(v + w) = Mv +

Mw and M(av) = aMv. These formulas follow from the distributive

and associative laws of matrix multiplication.

Conversely, if you have any permutation s of k

n

that always

satisﬁes the formulas s(v + w) = s(v) + s(w)ands(av) = as(v), then

there is some matrix M so that s is the same as multiplication by

M: s(v) = Mv for every v in k

n

.Ofcourse,M depends on s.

So one way of constructing linear representations of a group G is

by ﬁnding a vector space k

n

and a linear action of G on it. This is

by deﬁnition a morphism f from G to the permutation group of k

n

that satisﬁes the formulas s(v + w) = s(v) + s(w)ands(av) = as(v)

for every permutation s = f (g) coming from any group element

g in G.

For example, think about the group A

4

of symmetries of a

tetrahedron which we discussed in chapter 12. There we deﬁned

amorphismr with the property that for any element σ in A

4

,

r(σ ) is the rotation of space that causes the permutation σ on the

vertices of the tetrahedron. You can think of a rotation of space as

a permutation of R

3

. As such, it is a linear action on R

3

because

its effect as a permutation of R

3

can be given as multiplication

by a certain matrix in GL(3, R). (The particular matrix depends, of

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