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258 CHAPTER 23
On a parallel track, we introduced concepts speciﬁc to number
theory: systems of Z-equations, the integers modulo a prime num-
ber, the Galois group of a polynomial, and the monarch of our whole
domain—the absolute Galois group of Q. We tried especially hard
to give you a feeling for what this group consists of: permutations
of the ﬁeld of algebraic numbers that “preserve” addition and
multiplication, and hence permute the algebraic solutions of any
given system of Z-polynomial equations.
With these basic ideas under our belts, we proceeded to the
subject matter proper: the matrix representations of the absolute
Galois group of Q and how these are used to state (and even-
tually prove) reciprocity laws. We gave examples stemming from
quadratic reciprocity, roots of unity, and torsion points on elliptic
curves.
We also tried to explain a bigger picture where, by necessity,
we were vague. This involved more advanced concepts, such as
Q
p
,
´
etale cohomology, and generalized reciprocity laws. We can
only hope we were able to give you some intimation of what is
involved here, like a physicist trying to explain string theory to
the general public. The physicist, however, has the intuitively clear
physical analogies of strings and particles to go on, and we have
only mountains of abstractions to climb.
Back to Solving Equations
Along the way, we kept telling you that the guiding problem in this
branch of number theory is the study of solution sets of systems of
Z-equations in various number systems. Now we should say a little
more about how generalized reciprocity laws help in this project.
Which systems of Z-equations would we like to solve? They are
likely to be elegant or historical. For example, x
13
+ y
13
= z
13
is
both. It is an instance of Fermat’s equation, the one Fermat’s Last
Theorem refers to, and it is nicely symmetric and simple. As we
have remarked several times in this book, Wiles’s proof of Fermat’s
Last Theorem uses representations of the absolute Galois group of
Q in a central way.
RETROSPECT 259
To be speciﬁc, we remember that Fermat’s Last Theorem states
that if n is an integer greater than 2, then the Z-equation
x
n
+ y
n
= z
n
has no solution where x, y, and z are all integers, except
for the trivial solutions where one or more of the variables is set
equal to 0.
We can summarize chapter 22 as follows: First, we show that
it sufﬁces to assume that n is an odd prime p. Now suppose
you have nonzero integers a, b, and c and an odd prime p with
a
p
+ b
p
= c
p
. You can then construct an elliptic curve E (using this
hypothetical nontrivial solution of course) in a clever way
1
so that
the elliptic curve has the following property: The two-dimensional
representation φ of the absolute Galois group of Q on the -torsion
points of E is such that φ should enter into a reciprocity law with
a modular form f of level 2 and weight 2. You then compute that
a modular form with the properties that f must have cannot exist.
The last step—the nonexistence of f —is the easy part. The hard
part is establishing the reciprocity law.
Similar equations, of the form x
r
+ y
s
= z
t
, can also be approached
in this way. Besides the examples mentioned in chapter 22, Jordan
Ellenberg has shown, by proving and then using some new gener-
alized reciprocity laws, that:
THEOREM 23.1: Suppose x, y, and z are all nonzero integers
with no common factor, and p is a prime larger than 211.
Then it is impossible to have
x
4
+ y
2
= z
p
.
Here is an even more complicated example of reciprocity: Let S
be the Z-variety deﬁned by the equation in the ﬁve variables X, Y,
Z, W, and T:
X
5
+ Y
5
+ Z
5
+ W
5
+ T
5
5(XYZWT) = 0.
It has been proven that there is a modular form f whose Fourier
coefﬁcients enter into a reciprocity law that enables you to tell, for
1
In fact, E is given by the equation y
2
= x(x a
p
)(x + b
p
).

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