QUADRATIC RECIPROCITY REVISITED 217

Simultaneous Eigenelements

Let us return to one-dimensional representations, and show how

we can use the concept of a generalized reciprocity law to deduce

quadratic reciprocity. The black box associated with the Galois

representations we consider in this chapter is constructed from

an integer N > 1 and a set of functions F

N

with some special

properties. First we have to make the following deﬁnition:

DEFINITION: Let N be an integer greater than 1. Then

(Z/NZ)

×

is the set of all integers b modulo N that have

multiplicative inverses modulo N.

For example, 3 is in (Z/10Z)

×

because 3 · 7 ≡ 1 (mod 10), but 5

is not in (Z/10Z)

×

because there is no solution to the congruence

equation 5x ≡ 1 (mod 10). If b is in (Z/NZ)

×

, we write b

−1

for the

element in (Z/NZ)

×

that satisﬁes the congruence bb

−1

≡ 1 (mod N).

A theorem that is proved in most introductory number the-

ory courses says that the elements of (Z/NZ)

×

are the numbers

between 1 and N − 1 that have no factor in common with N other

than 1.

Now we can deﬁne the set F

N

:

DEFINITION: Let N be an integer greater than 1. Then F

N

stands for the set of all functions from (Z/NZ)

×

→ C

×

.

1

If α is a function in F

N

, and if b is an element of (Z/NZ)

×

, then

as usual with functional notation, we write α(b) for the complex

number in C

×

that the rule α assigns to b.

The set F

N

contains the black box. To describe these black boxes,

we need to consider certain functions inside of this set.

DEFINITION: Let N be an integer greater than 1. A

simultaneous eigenelement in F

N

is a function α in F

N

with

1

Recall that C

×

denotes the set of all nonzero complex numbers.

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