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Fearless Symmetry by Robert Gross, Avner Ash

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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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