RECIPROCITY LAWS 195

Black Boxes

Whatever the independent way of generating the χ

φ

(Frob

p

)’s might

be, we can think of it as a black box.

1

It is usually some complicated

object from some other part of mathematics. In this book we view

some of these mathematical machines that produce the sequence of

numbers χ

φ

(Frob

p

)’s as a given, the way a child can read the dial

of a wristwatch and tell the time without knowing how the watch

works. In some cases (e.g., quadratic reciprocity), we will be able to

describe completely what is happening.

From the point of view we take in this book, a reciprocity law is

a black box. You put in a prime p (where φ is unramiﬁed) and out

pops a number. If it is the black box with the label φ on it, then

that number will be χ

φ

(Frob

p

). The possession of this black box then

gives us power over the list of χ

φ

(Frob

p

)’s and over whatever it is in

number theory that they tell us about, for example, torsion points

on some elliptic curve, or indirectly about a supposed solution to

Fermat’s equation x

n

+ y

n

= z

n

.

You may suppose that there should be exactly one black box for

each φ, and it should be labeled φ. Then if you want information

about the sequence of numbers χ

φ

(Frob

p

), you just pull out the box

labeled φ and start throwing in the p’s.

Things are not so simple. There can be many black boxes that

all give the same sequence, so they all belong to φ. They might be

of different types (e.g., some modular forms and some cohomology

classes—see chapters 20 and 21 for these concepts). Also, the black

boxes do not usually come labeled by the φ’s. Usually they are la-

beled with information that is derivable from some properties of φ.

This works best if there is only a ﬁnite number of black boxes in our

inventory with any particular label.

To summarize: The equality between the traces of the matrices

in a Galois representation and numbers produced by some sort of

black box is what is called a reciprocity law. There are various black

1

A black box is an input–output box that you cannot see into. It may have a label on the

outside, and you know what it does to the input, but you do not have to understand why

it works. This use of the term has nothing to do with the “black box” that records ﬂight

data on an airplane.

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