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Fearless Symmetry by Robert Gross, Avner Ash

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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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