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Fearless Symmetry by Robert Gross, Avner Ash

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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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