ONE- AND TWO-DIMENSIONAL REPRESENTATIONS 205

In other words, you can always ﬁgure out what φ(γ )isby

applying γ to ζ : φ(γ ) = a if and only if γ (ζ ) = ζ

a

, or, concisely,

γ (ζ ) = ζ

φ(γ )

.

Using this formula, you can easily check that φ is a morphism.

Say φ(γ ) = a and φ(γ

) = a

. Then what is φ(γ ◦ γ

)? Use our test,

and apply γ ◦ γ

to ζ : γ ◦ γ

(ζ ) = γ (γ

(ζ )) = γ (ζ

a

) = γ (ζ )

a

(since γ

respects all arithmetic operations, including raising to a power)

= (ζ

a

)

a

= ζ

aa

by the laws of exponents. So the test tells us that

φ(γ ◦ γ

) = aa

. But this is equal to φ(γ )φ(γ

). So φ(γ ◦ γ

) =

φ(γ )φ(γ

) for all elements γ and γ

in G—and that is the deﬁning

property of a morphism.

So φ really is a representation of G. It is a linear representation

to GL(1, F

p

). Because we are using only 1-by-1 matrices, we call

it a “one-dimensional” representation. So χ

φ

(Frob

q

) is the same as

φ(Frob

q

).

5

Finally, a prime q is unramiﬁed in Q(x

p

− 1) as long as

q = p, and the reciprocity law that we get from Theorem 18.2 says

that

χ

φ

(Frob

q

) = φ(Frob

q

) ≡ q (mod p).

This is a brilliant example of a reciprocity law. The black box labeled

φ simply outputs q when you input q.

Two-Dimensional Galois Representations Arising from

the p-Torsion Points of an Elliptic Curve

We now switch our attention to elliptic curves. Pick an elliptic curve

E, and let p be some prime. We have written earlier about the

set E[p], the p-torsion of the elliptic curve. These are the points

P on E(C) that solve the equation pP = O. (This equation should be

thought of as similar to the equation x

p

= 1 above.) Remember that

the set E[p] has a structure: We can ﬁnd two particular elements

P and Q in this set so that every element in E[p] can be written as

aP + bQ, where the numbers a and b are between 0 and p − 1(and

5

Remember that the trace of the 1-by-1 matrix

"

q

#

is just the number q itself.

206 CHAPTER 18

so we can think of them as element of F

p

). If σ is any element of G,

we can associate a matrix to σ as on page 147; we will now call that

matrix ψ(σ ).

We claim that ψ is a representation of G, this time to GL(2, F

p

).

If σ is any element in the absolute Galois group G, ψ(σ ) depends

only on the restriction of σ to the ﬁeld Q(E[p]) generated by the

coordinates of the p-torsion points of the elliptic curve E. So you can

see what 2-by-2 matrix ψ sends σ to by applying σ to P and to Q and

writing down the answers in terms of P and Q.Ifσ (P) = aP + bQ

and σ (Q) = cP + dQ, then ψ(σ ) is the matrix

ac

bd

.

Again, you can check that ψ is a morphism. It is a little more

complicated, involving multiplication of two 2-by-2 matrices. When

you try to check this, remember that applying σ ◦ σ

to P and Q

means ﬁrst applying σ

and then applying σ .

So ψ is a linear representation of G to GL(2, F

p

). It is a two-

dimensional linear Galois representation.

In the case of the one-dimensional linear representation φ above,

we saw that φ(Frob

q

) tells us something very interesting about the

prime q, namely, it tells us how the polynomial x

p

− 1 factors over

F

q

, which further tells us things such as whether F

q

contains a

primitive pth-root of unity. Similarly, ψ tells us some very inter-

esting things about the elliptic curve E. For example, if h(x)isthe

Z-polynomial whose roots are the ﬁrst coordinates of all the p-

torsion points on E, then ψ(Frob

q

) gives information about how h(x)

factors modulo q.

But there is much more to know about an elliptic curve. The

Galois representation ψ , and similar (but more complicated) con-

structions that use all the torsion points of E(Q

alg

), can tell us about

that extra information. For example, they can tell us whether E

has extra symmetries of its own, called “complex multiplications.”

Also, they can be used to make predictions about whether there

are inﬁnitely many rational points in E(Q), summed up in what

is known as the Birch–Swinnerton-Dyer Conjecture, named after

the two number theorists Bryan Birch and Peter Swinnerton-Dyer.

These things are unfortunately too advanced to explain in this

book, but it is good to remember that no matter how difﬁcult or

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