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Fearless Symmetry by Robert Gross, Avner Ash

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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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