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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