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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