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Fearless Symmetry by Robert Gross, Avner Ash

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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.

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