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Fearless Symmetry by Robert Gross, Avner Ash

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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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