260 CHAPTER 23

each prime p not dividing the level, the number of solutions

in S(F

p

). For a survey of this and many similar examples, see

(Yui, 2003).

Digression: Why Do Math?

We can attempt an answer to the question “Why do math?” as

illustrated by the mathematics discussed in this book.

If mathematics is the study of patterns, then we immediately

see the two reasons throughout history why people have “done

math”:

1. Curiosity about patterns, particularly of numbers and

shapes, and an æsthetic appreciation of them.

2. The need to study patterns occurring in the “real world.”

People sufﬁciently motivated by the ﬁrst reason become “pure”

mathematicians. They generally cannot understand why everyone

does not share their enthusiasm for the beauty of these patterns.

People motivated by the second reason are everyone (including

also “pure” mathematicians). Even in mundane affairs, you cannot

ignore patterns, such as those in your checkbook or the number

patterns in the bus schedule; the arrangement of the rooms and

furniture in your apartment (spatial patterns); the knotting of

DNA (topological patterns) if you are a molecular biologist, “etc.

etc. etc.”

We see in these two reasons the sources of “pure” and “applied”

mathematics, respectively. This book has concerned itself with a

topic in pure mathematics, a subtopic of number theory. Number

theory is often thought of as the purest of mathematics. Yet even the

ideas in this book, such as prime numbers,

´

etale cohomology, and

the corresponding Galois representations, have been ﬁnding their

applications in the “real world.” For example, Andr

´

e Weil’s version

of the Riemann Hypothesis for curves over F

p

has been applied

to the problem of constructing efﬁcient communication networks.

The use of prime numbers and their theory in the construction

of public key codes is of ever-increasing importance for ﬁnancial

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