A LAST LOOK AT RECIPROCITY 239

Review of Reciprocity Laws

Here are the examples of reciprocity laws mentioned in this book:

•

Quadratic reciprocity.

•

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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