In this chapter we discuss the idea of a “generalized

reciprocity law” more broadly. We put it in the context of

mathematical patterns. Looking again at the examples of

general reciprocity that we described in earlier chapters,

and introducing some new examples, we attempt to give

the ﬂavor of the most general reciprocity laws currently

conjectured. They form part of what is commonly known

as the “Langlands program.”

What Is Mathematics?

A fair deﬁnition of mathematics might be “the logical study of

patterns”:

•

Patterns of numbers—look at the chapters on quadratic

reciprocity for some complicated but beautiful examples.

•

Patterns of permutations—for example, the behavior of

cycle decompositions.

•

Patterns of points—this is geometry.

•

Patterns of solutions to systems of Z-equations—this is

what this book is all about.

We use the term “pattern” in the broadest possible sense, to mean

any arrangement of things that follows some orderly rule, allowing

for prediction or contemplation.

234 CHAPTER 21

A simple pattern is the empty set.

1

If you know the solution to

some problem is the empty set, you may know a lot. For example,

“What is the set of all eternal truths?” We are not saying the

solution is the empty set, but if it is, it might be nice to know it.

Of course, this is not a mathematical problem.

“What are all possible ways of arranging your dinner guests

around a circular table so no one sits next to his or her spouse?”

Such patterns are the province of the ﬁeld of mathematics called

combinatorics, and are also very important in probability theory.

“What are all solutions to the Z-equation x

n

+ y

n

= z

n

, where x, y,

and z are positive integers and n is an integer greater than 2?” We

know now that the solution is the empty set. That is the conjecture

of Fermat, now proved by Wiles.

But how did he prove it? Roughly speaking, mathematicians

had already proved that if there were any solutions, there would

be some pattern—in fact, a certain two-dimensional linear repre-

sentation of the absolute Galois group G. Wiles showed that this

particular pattern could not exist. In fact he proved something

much more important—a certain reciprocity law connected with

elliptic curves and modular forms.

2

That reciprocity law then

had the corollary that the Galois representation coming from a

supposed solution of x

n

+ y

n

= z

n

could not exist. We will explain

this in more detail in chapter 22.

So, sometimes, even to show that a certain pattern is very simple

(e.g., empty), you have to know about what other sorts of patterns

can exist. Two basic types of problems in mathematics can be

phrased as follows:

1. Can a pattern with certain given properties exist or not?

2. Classify all patterns with the given properties.

The ﬁrst statement can be called “the existence question” and the

second statement the “classiﬁcation problem.”

The reason mathematics has so many applications to other ﬁelds,

such as physics, chemistry, biology, economics, and so on, is that

1

Perhaps only a mathematician would think of the empty set as a pattern—the “null

pattern.”

2

See later in this chapter for a discussion of modular forms.

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