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Fearless Symmetry by Robert Gross, Avner Ash

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Starting in this chapter we usually use lowercase Greek
letters for Galois representations.
Suppose that we have a linear representation φ of the
absolute Galois group G. We will produce a list of numbers
by taking the trace of φ(Frob
2
), φ(Frob
3
), ..., one number
for each prime that is unramified for the representation φ.
This list encodes a lot of crucial information about G and φ
which can be used to study Z-equations. The information
is summarized in what are called generalized reciprocity
laws.
In this chapter, we explain the general concept of a
reciprocity law. In the rest of the book, we will give
examples of reciprocity laws and how they work to provide
information about solutions to Z-equations. In chapter 19,
we will reinterpret quadratic reciprocity as a reciprocity
law in this new, generalized, sense. The connection with
quadratic reciprocity is the reason we call theorems of this
type reciprocity laws.
The List of Traces of Frobenius
A little review: We have the set of all algebraic numbers, denoted
Q
alg
. We have the absolute Galois group G of all arithmetic-
preserving permutations of Q
alg
.IfR is some number system, n 1
194 CHAPTER 17
is some integer and φ : G GL(n, R) is a morphism of groups, then
we say that φ is a linear Galois representation with coefficients
in R.
We have told you that each φ comes with a set of “bad primes,
the “ramified primes. In all of the examples we will consider, this
set of bad primes is finite. If p is not in this finite list of bad primes,
we say that φ is unramified at p.
If φ is such a linear Galois representation, and σ is any element
in G, then we have defined the character of φ at σ to be the value of
the trace of φ(σ ). Recall what this means: φ(σ )isann-by-n matrix
with entries in R, and its trace is the sum of the entries down
the main diagonal, going from the upper-left to the lower-right. In
symbols, we write χ
φ
(σ ) for the value of this character at σ .Itisa
number in R.
We also have defined certain special elements of G, the Frobenius
elements. They are not really single elements, but rather certain
sets of elements of G, one for each prime number p.Onekey
property they share is the following: If φ is unramified at p, then
χ
φ
(Frob
p
) is a well-defined element of R. This means that it does
not matter which σ you choose from the whole set of possibilities
for Frob
p
; χ
φ
(σ ) is always the same number, and we call the common
value χ
φ
(Frob
p
).
In the next few chapters we hope to show by example that this
list of numbers χ
φ
(Frob
2
), χ
φ
(Frob
3
), χ
φ
(Frob
5
), ... (omitting from
our list those p where φ is ramified, in which case χ
φ
(Frob
p
) is not
really defined), is always a very interesting list of numbers. The
whole idea of reciprocity laws is to try to find other independent
ways of generating these lists of numbers. If we succeed, we obtain
a type of theorem called a generalized reciprocity law, or simply a
reciprocity law for short.
We will be able to prove for you that a particular reciprocity law
is true in only a few cases. Usually, these proofs are extremely dif-
ficult and require much additional mathematics that lies outside of
the scope of this book. For the most general reciprocity laws, proofs
have not yet been discovered, although the experts conjecture that
the laws are true.

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