A MACHINE FOR MAKING GALOIS REPRESENTATIONS 231

these ﬁelds were ﬁrst discovered by Galois. If m = 1, we get back

our old friend F

q

.

Now let W be a Z-variety. We can then look at the solutions

of the equations deﬁning W where we substitute elements from

F

q

m

. In other words we look at W(F

q

m

). It turns out that the set

of numbers #W(F

q

m

) is closely connected to the matrices giving the

action of Frob

q

on H

j

(W, Q

p

)asweletj = 0, 1, 2, ....Thisfactisa

vast generalization of equation (20.1) for elliptic curves. It gives us

a concrete link between the

´

etale cohomology of W and solving the

Z-equations that deﬁne W.

You may object that this link only goes to solutions where the

variables are set equal to elements of these weird ﬁnite ﬁelds,

W(F

q

m

). What about the solutions we are really interested in, that

is to say, what about W(Q)? This is one of the main current topics

of research. Suppose we ask: Is W(Q) ﬁnite or inﬁnite? Current

research suggests looking at #W(F

q

m

)forall q and m.Ifthis

number tends to be bigger than average (in a certain precise,

technical sense) then you can conjecture that it is because W(Q)

has lots of elements, and their reduction modulo q for all the

different q’s is what accounts for the bigger than average size of the

#W(F

q

m

)’s. In the case of elliptic curves, this statement is called the

Birch–Swinnerton-Dyer Conjecture.

End of skippable detail.

Conjectures about

´

Etale Cohomology

The

´

etale cohomology of Z-varieties furnishes an abundance of

linear representations of G. These representations all share certain

technical properties that make it possible to work with them,

although very much about them is still unknown. The conjecture

of Jean-Marc Fontaine and Barry Mazur states that any linear

representation of G over Q

p

that shares these technical properties

actually results as part of an

´

etale cohomology group of some

Z-variety W. This is a powerful conjecture that is very far from

having been proved. It is safe to say that no one has any idea how

to prove it in general.

232 CHAPTER 20

You can see how the Fontaine–Mazur Conjecture may have been

made: You think of every way possible to do something—in this

case ﬁnding linear Galois representations that satisfy the nice

properties—and then you conjecture that there are no other ways.

If you are smart enough, your conjecture may hold true, because

you are so smart and you were not able to think of any other wa ys

of doing it.

Another way of making conjectures is to have the intuition

that some analogy should hold. This is how

´

etale cohomology was

discovered. Very roughly speaking, Andr

´

e Weil conjectured that

some such cohomology theory must exist, based on the analogy from

topology. Then Alexander Grothendieck ﬁgured out a construction,

still based on the topological analogy, that gave the theory a reality.

Then Pierre Deligne proved many of its most important properties.

This all happened in the middle of the last century, over a period of

20 or 30 years.

At the moment,

´

etale cohomology, and some other kinds of

similar cohomology theories, are the most general known ways of

constructing “nice” linear representations of the absolute Galois

group G of Q. It seems that further study of these cohomology

theories, in tandem with a deepening knowledge of G, will be the

most likely direction in which the study of generalized reciprocity

in number theory will go. We expect there to be beautiful reci-

procity laws involving the action of Frobenius elements on

´

etale

cohomology groups and modular forms (see chapter 21) and their

generalizations, called “automorphic representations .” (Yes—more

representations!)

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