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Fearless Symmetry by Robert Gross, Avner Ash

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QUADRATIC RECIPROCITY REVISITED 221
α(py) = a
1
p
α(y). We know that a
1
p
= α(p), so make that one final
substitution, and we get the equation α(py) = α(p)α(y). If x is any
number in (Z/NZ)
×
, we can apply this formula one by one to all of
the prime factors of x, and conclude that α(xy) = α(x)α(y).
In other words, the eigenelement α is a morphism of (Z/NZ)
×
to
C
×
, that is,
α(xy) = α(x)α(y).
Moreover, if α is an eigenelement from Theorem 19.3 (for some
W), α(p) = a
1
p
= χ(Frob
p
)
1
= χ(Frob
p
) because χ (Frob
p
) = 1or1.
Note also that a
1
p
= a
p
for the same reason.
Summarize all of this as the following:
THEOREM 19.4: If α is a simultaneous eigenelement of F
N
satisfying Theorem 19.3 such that α(1) = 1, then α is a
morphism from (Z/NZ)
×
to C
×
, and the eigenvalues are given
by the formula
a
p
= α(p) = χ(Frob
p
) =
W
p
.
The last two equalities follow from Theorems 19.2 and 19.3
above. In particular we have the formula
α(p) =
W
p
which we will use freely from now on.
A Strong Reciprocity Law
To be able to derive quadratic reciprocity, we need to strengthen our
Weak Reciprocity Law. Given W, we have to tell you what N to use.
THEOREM 19.5 (Strong Version of the Reciprocity Law):
In Theorem 19.3, we can let N = 4
|
W
|
.

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