196 CHAPTER 17

boxes that can be used, which generally have something to do with

geometry or topology. The use of these black boxes (which may be

more or less transparent to the number theorist who is using them)

is at the heart of the method of Galois representations in modern

number theory.

Weak and Strong Reciprocity Laws

In a reciprocity law, we ﬁrst specify the kind of black box we are

going to consider. For instance, we specify elliptic curves. Then,

for each kind of black box, there are two kinds of reciprocity law:

weak and strong. In a weak reciprocity law, all we are told is that

for every Galois representation φ of a certain type, there is some

black box that outputs the corresponding χ

φ

(Frob

p

)’s. In the strong

reciprocity law, we are also told the label on the black box. If we

have not proved a reciprocity law yet, but are only guessing it is

true, then we call these two different types of statements the “weak

conjecture” and the “strong conjecture.”

You can see that it is much easier to verify or disprove a strong

conjecture, because knowledge of the label cuts down to a ﬁnite

amount the number of black boxes for which you have to look. If

true, the strong reciprocity law gives more information than the

corresponding weak one.

It is obviously better to try to prove the strong conjecture if

possible. In applications such as Wiles’s proof of Fermat’s Last

Theorem, it was necessary to prove a strong reciprocity law to ﬁnish

the proof. In terms of our analogy, the proof went like this: Suppose

you have two nonzero nth powers that add up to another nth

power. From this equation, you can deduce the existence of a certain

Galois representation. The strong reciprocity law you have already

proved implies the existence of a black box with a certain label.

You work out the label of the black box. You go to your inventory

of those black boxes—and there are not any with that label!

Contradiction.

In summary, a strong reciprocity law tells us that the list of

numbers coming from a particular Galois representation, produced

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