220 CHAPTER 19

believable, because if you look back at the deﬁnition of χ(σ ), you

will see that χ (σ ) has to be 1 or −1.

A Weak Reciprocity Law

THEOREM 19.3 (A Weak Reciprocity Law): Given a

square-free integer W,

4

there is some positive integer N and a

simultaneous eigenelement α in F

N

with eigenvalues

a

2

, a

3

, ... such that

χ(Frob

p

) = a

p

for every prime p that is not a divisor of N. (If p is not a

divisor of N, then p is unramiﬁed for the Galois

representation χ.)

Why is this a reciprocity law? It tells us how to ﬁnd the list of

numbers χ (Frob

p

) in terms of the black box α. It does not tell us

exactly what α is (although we will be able to ﬁgure that out later),

but the assertion of the mere existence of α gets us going. We will

need to strengthen this law a bit, and then it will imply quadratic

reciprocity.

To see how this reciprocity law implies quadratic reciprocity, we

ﬁrst have to discover what the eigenvalues are for a simultaneous

eigenelement α in F

N

. So let us suppose that α is a simultaneous

eigenelement. We may multiply α by any constant in C

×

and it

will still satisfy equation (19.1), with the same eigenvalues. If we

multiply α by a constant c which has the property that cα(1) = 1,

we get a function that sends 1 to 1. In other words, henceforth we

can assume that α(1) = 1.

From our deﬁnitions, we know that α(p

−1

b) = a

p

α(b) for any

number b in (Z/NZ)

×

. Set b = p, and we have α(p) = a

−1

p

. Next,

take the formula α(p

−1

b) = a

p

α(b), and let b = py. We get α(y) =

a

p

α(py). Move a

p

to the other side of the equation, and we have

4

A square-free integer is one that is not divisible by any perfect square other than 1. For

example, 30 is square-free, and 12 is not square-free.

Start Free Trial

No credit card required