218 CHAPTER 19

the property that for each prime p that is not a divisor of N,

there is a complex number a

p

such that

α(p

−1

x) = a

p

α(x), (19.1)

if x is any element of (Z/NZ)

×

. (Note that p

−1

x is also in

(Z/NZ)

×

, because of the assumption that p is not a divisor

of N.) The numbers a

2

, a

3

, ... are called the eigenvalues

corresponding to the function α.

An equation similar to (19.1) turns out to be true in many

generalized reciprocity laws. Unfortunately, the equation is usually

more complicated than this one.

Equation (19.1) has surprising consequences which we will

explore in the remainder of this chapter. We can start here with an

observation. Suppose that p and q are two primes that satisfy the

congruence p ≡ q (mod N). Then p

−1

≡ q

−1

(mod N), and therefore

if x is any integer at all, p

−1

x ≡ q

−1

x (mod N). This means that

α(p

−1

x) = α(q

−1

x), because α is a function from (Z/NZ)

×

. The right-

hand side of equation (19.1) now tells us that a

p

α(x) = a

q

α(x), and,

because α(x) is nonzero, we can conclude that a

p

= a

q

.

This is worth restating: If p ≡ q (mod N), then a

p

= a

q

. This

equality is a special property of eigenvalues that occur in the study

of one-dimensional Galois representations.

The Z-Variety x

2

− W

What do these eigenvalues mean?

A reciprocity law would tell us that this list a

2

, a

3

, ... should

have something to do with the traces of Frobenius elements Frob

2

,

Frob

3

, ... under some Galois representation φ. It turns out that

this is true, and that we get the correct φ by studying the Z-variety

x

2

− W = 0 for some nonzero integer W. We assume that W is not a

perfect square. Then any element σ in the absolute Galois group

G must take a solution of x

2

− W = 0 to a solution of the same

equation. We designate one of the solutions as

√

W, and then the

other is −

√

W. Hence, σ (

√

W) =

√

W or σ (

√

W) =−

√

W. This allows

QUADRATIC RECIPROCITY REVISITED 219

us to deﬁne a function φ from G to {+1, −1} by the rule

σ (

√

W) = φ(σ )

√

W.

You can easily check that φ is a morphism from G to {+1, −1}.

2

Because +1 and −1 are numbers, we can view φ as a Galois

representation from G to GL(1, C), viewing numbers as being the

same as 1-by-1 matrices, as we have done before.

Because the trace of a 1-by-1 matrix is the same as the number

in the matrix, we see that the character χ

φ

of the representation φ

is just φ. To emphasize this, we will use the symbol χ rather than φ.

To summarize: For any nonsquare integer W, we have the Galois

representation χ from G to GL(1, C) deﬁned by

σ (

√

W) = χ(σ )

√

W.

We really should include the letter W somewhere in this notation,

because χ depends on W, but the notation χ

(W)

is too cluttered.

From the deﬁnition of Frob

p

, or from the fact that the cycle type

of χ(Frob

p

) is related to how x

2

− W factors modulo p,

3

follows this

next fact:

THEOREM 19.2: If p is an odd prime that does not divide W,

p is unramiﬁed for χ and

χ(Frob

p

) =

W

p

.

Remember that

W

p

is called the Legendre symbol. It does not

refer to dividing W by p. The equation in the theorem is at least

2

The group law on {+1, −1} is multiplication.

3

The set of elements Frob

p

in the absolute Galois group of Q gives information about

how polynomials factor in F

p

and vice versa. If

W

p

= 1, then the congruence y

2

≡ W

(mod p) has two solutions, say a and b. Then the polynomial x

2

− W factors as x

2

− W ≡

(x − a)(x − b) (mod p). On the other hand, if

W

p

=−1, then the congruence y

2

≡ W

(mod p) has no solutions, and the polynomial x

2

− W does not factor into the product of

two linear polynomials modulo p. Now, by Theorem 16.1, in the ﬁrst case, the cycle type

of χ(Frob

p

)is1+ 1sothatχ (Frob

p

) must be the neutral element 1 in the multiplicative

group {+1, −1}. In the second case, the cycle type of χ (Frob

p

) is 2 so that χ(Frob

p

) must

be the other element −1 in the group.

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