A MACHINE FOR MAKING GALOIS REPRESENTATIONS 229

be souped up to study all kinds of motion governed by differential

equations, as studied in calculus.

The idea o f linearization has also been used by topologists.

Consider a doughnut sitting in space, or more precisely the surface

of the doughnut. Suppose it is a very smooth surface, say an old-

fashioned without frosting. This is a somewhat wavy surface and

hard to study exactly, because the equations that describe it are

very complicated. Topologists deﬁned the tangent plane at each

point of the surface. Each tangent plane is nice and linear, and

some problems about the surface can be translated into questions

of linear representations of a certain group (a monodromy group)

on these tangent planes. Then topologists went even further and

ﬁgured out how to linearize some of the topological structure of

surfaces, for example, how many holes they have. The vector spaces

they invented are called “cohomology groups.” A doughnut has only

one hole, but a pretzel can have more than one.

Although it is not difﬁcult to count the holes in a real pretzel in

your hand, prior to eating it, when a surface pops out of an abstract

mathematical construction it can be very difﬁcult to ﬁgure out its

properties, such as how many holes it has. The cohomology groups

canhelpustodoso.

´

Etale Cohomology

Algebraic geometers were able to make an amazing translation of

these topological ideas to varieties, coming up with

´

etale cohomol-

ogy. Let W be a Z-variety. Choose a prime number p.Thereisaﬁeld

called Q

p

, too complicated to be deﬁned here.

1

It is a sort of cross

between F

p

and Q. For each non-negative integer j, there is a set of

vectors deﬁned over Q

p

denoted H

j

(W, Q

p

), which is called the jth

´

etale cohomology group of W. It comes with a linear action of G,the

absolute Galois group of Q. It is related algebraically to W in a way

1

If you know how to write numbers in base p, you can deﬁne Q

p

as the set of all

“inﬁnite p-adic expansions” ···a

3

a

2

a

1

a

0

.b

1

b

2

b

3

···b

r

. These are like decimal expansions

of numbers in R, except that they are allowed to be inﬁnite to the left but not to the right,

and are in base p instead of base 10. We leave it to you to ﬁgure out how to add and

multiply them, or you can see (Koblitz, 1984).

230 CHAPTER 20

that is similar to the way that topological cohomology groups are

related to topological surfaces.

For example, if E is an elliptic curve, then H

1

(E, Q

p

)isvery

closely related to the mod p linear representation of Q studied in

chapter 18. What we considered in that chapter corresponds only to

the mod p version of this

´

etale cohomology, namely, H

1

(E, F

p

), but

the whole construction can be lifted up from F

p

to Q

p

.

To give you some idea of what Q

p

is, we can say this. Replacing

F

p

by Q

p

turns out to be equivalent, in the case of elliptic curves, to

considering all the p

m

-torsion points on E,form = 1, 2, 3, ....Ifwe

do this, we can generalize the formula

χ

ψ

(Frob

q

) ≡ 1 + q − #E(F

q

)(modp). (20.1)

from chapter 18 in such a way that it determines #E(F

q

)exactly,

not just modulo p. This is related to the fact that if there is some

integer x and you know what x is modulo p

m

for all m,thenyoucan

ﬁgure out what x is exactly (by taking m large enough.)

One nifty thing is that the same Z-variety W has an entire

sequence of

´

etale cohomology groups as p runs through all the pos-

sible prime numbers . But the basic number theory, for instance, the

trace of Frob

q

for unramiﬁed primes q, will be “independent” of p in

the following sense. The trace of Frob

q

in the representation of G

acting on H

j

(W, Q

p

) is a number in Q

p

. Because the ﬁelds Q

p

are all

different for different p’s, it seems as if we could not compare these

traces as p varies. But it turns out that the trace of Frob

q

is always

an algebraic integer—it is a root of some polynomial f

p

(x)withZ-

coefﬁcients and leading coefﬁcient 1. (We should have told you that

Z is a subset of Q

p

for every p.) So we can compare the traces by

comparing the polynomials f

p

(x). If we take care to use the polyno-

mial of smallest possible degree, we will see that it is alwa ys the

same, independently of which p we are using. W e call the set of Ga-

lois representations on H

j

(W, Q

p

)asp varies a “compatible family.”

Here is some skippable detail about

´

etale cohomology. Let q be a

prime number, which we will assume to be unramiﬁed in the Galois

representations we consider. For any integer m ≥ 1, there is a ﬁeld

with exactly q

m

elements, called F

q

m

. This is one of the ﬁrst things

you prove in an undergraduate course in Galois theory, and in fact

Start Free Trial

No credit card required