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A LAST LOOK AT RECIPROCITY 239
Review of Reciprocity Laws
Here are the examples of reciprocity laws mentioned in this book:
Quadratic reciprocity viewed as reciprocity between Frob
in a Galois representation and the factorization of X
2
W
modulo .
Reciprocity between Frob
in a Galois representation and
the factorization of a general irreducible Z-polynomial
modulo .
Reciprocity between Frob
in a Galois representation and
the solution set of an elliptic curve over F
.
Reciprocity between Frob
in a certain kind of
two-dimensional Galois representation and a modular
form.
From this point of view, the big theorem that implied Fermat’s
Last Theorem, and which was proved in several papers by Wiles
and his colleagues, established a reciprocity law: For any elliptic
curve E with coefﬁcients in Q there is some modular form q + a
2
q
2
+
a
3
q
3
+··· with the property that for any prime (except for a ﬁnite
number of exceptions) a
= 1 + #E(F
).
8
We can give a numerical example of this last kind of reci-
procity. Let E be the elliptic curve deﬁned by the equation y
2
+ y =
x
3
x
2
. (We could write an equation for E in the usual form as
y
2
= x
3
432x + 8, 208, but by using the equation y
2
+ y = x
3
x
2
,
we keep the coefﬁcients a lot smaller and avoid technical prob-
lems computing a
2
and a
3
.) The only “bad” prime for E is 11,
and for any prime other than 11, we can deﬁne a
by the
formula
a
= 1 + #E(F
).
For example, E(F
2
) ={O, (0, 0), (0, 1), (1, 0), (1, 1)}, and so a
2
= 1 +
2 5 =−2. Similarly, E(F
3
) ={O, (0, 0), (0, 2), (1, 0), (1, 2)}, and so
8
This is a reciprocity law between the modular form and the Galois representations
coming from the torsion points on E.

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