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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