228 CHAPTER 20

course, on the particular rotation.) This gives a linear action of A

4

on R

3

:Foranyσ in A

4

, r(σ )actsonR

3

by rotating it.

Linearization

We have seen that G, the absolute Galois group of Q,hasa

permutation representation on the roots of any given Z-polynomial.

Similarly, G has a permutation representation on the Q

alg

-points of

any Z-variety W, deﬁned by permuting all of the solutions in Q

alg

of

the system of equations deﬁning W. This works ﬁne when studying

Z-varieties deﬁned by a single polynomial in a single variable.

But when the varieties get more complicated, these permutation

representations are very hard to work with. We want to relate them

to some linear representations of G to get a better handle on the

number-theoretic properties of the variety W.

What is needed is a way to replace the permutation representa-

tion of G on the Q

alg

points of W by a related linear representation

of G on some set of vectors associated with W.Thisiswhat

´

etale

cohomology does. The process of replacing a complicated object by a

simpler linear object is called linearization.Itiswhatmustbedone

to Z-varieties in order to get the Galois representations we want.

For a much simpler example of this process of linearization,

consider the old problem of ﬁguring out your momentary speed

while driving along a highway. If you are driving at a constant speed

c, and you graph your distance driven versus the time, you will get

a straight line with slope c. So the slope is your speed. But if you are

speeding up and slowing down, the graph will be a curve. What is

your speed exactly now (say at t = 10.2 seconds after you started)?

To answer this question you have to replace the curved graph

by the straight line that approximates it most closely just at time

t = 10.2. This is the tangent line to the graph at that point. It has a

slope, and that slope is your instantaneous speed at time 10.2.

We say that we have linearized the graph of distance versus

time at that point by replacing it with the tangent line. If you are

interested only in your motion in a very small time frame around

t = 10.2, then this line will give a good approximation. This idea can

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