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Fearless Symmetry by Robert Gross, Avner Ash

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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 qs 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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