ONE- AND TWO-DIMENSIONAL REPRESENTATIONS 207

deep is something we do discuss, there are plenty of more difﬁcult

and much deeper facts awaiting discussion. For a little more about

this, see chapters 21 and 22.

How Frob

q

Acts on p-Torsion Points

Note that because we made some choices, namely, of P and Q, the

matrix ψ(σ ) could change if our choices changed, but it will stay in

the same conjugacy class. Hence, as we know from chapter 16, the

trace of this matrix will not change. In other words, the character

χ

ψ

(σ ) is an element of F

p

that can be computed and will stay the

same, regardless of how we choose P and Q.

Here now is an amazing theorem connecting χ

ψ

(Frob

q

)andE(F

q

),

the set of solutions modulo q of the equation y

2

= x

3

+ Ax + B that

deﬁnes the elliptic curve E (plus the point O as usual):

THEOREM 18.5: Let q be a prime other than p that is

unramiﬁed for the Galois representation ψ. (This will be true

if q does not divide 2p(4A

3

+ 27B

2

).) The matrix ψ(Frob

q

)is

only deﬁned up to conjugacy, but χ

ψ

(Frob

q

) is well-deﬁned,

and

χ

ψ

(Frob

q

) ≡ 1 + q − #E(F

q

) (mod p).

Here #E(F

q

) means the number of points in E(F

q

). In other

words, #E(F

q

) equals the number of solutions to the

congruence y

2

≡ x

3

+ Ax + B (mod q) deﬁning the elliptic

curve E, plus 1 (for O, the point at inﬁnity).

Now we are keeping track of the rather subtle information of

whether or not x

3

+ Ax + B is a square modulo q! The formula

above is both strikingly interesting and only the tip of an iceberg.

There are similar formulas, although much more complicated, for

all Z-varieties. Consider what this formula means in the case of an

elliptic curve. Given the elliptic curve, we get this two-dimensional

Galois representation ψ. It has to do with the p-torsion on the

elliptic curve. From ψ, we take only its trace, χ

ψ

. When we apply

it to Frob

q

we get a number closely related to the number of

208 CHAPTER 18

mod q solutions to the cubic equation that deﬁnes the elliptic

curve!

We use the elliptic curve E and the prime p to compute a machine

(the Galois representation), put Frob

q

through the machine, and

get out an interesting fact about q. We could just as well start

out with any other p (as long as it is not equal to q) and get the

“same” fact about q, because after all #E(F

q

) depends only on q,

not on p. (We put “same” in quotes because we are looking at the

same number 1 + q − #E(F

q

) but modulo different primes p.) It’s

funny: The Galois representation ψ does depend on p. But when we

apply χ

ψ

to Frob

q

we get the “same answer”: 1 + q − #E(F

q

) (mod p),

no matter what p is!

6

If we let p vary and for each p take the corresponding ψ , we get

what is called a compatible family of Galois representations, be-

cause they have the “same” traces on Frobenius elements. This is a

general fact about

´

etale cohomology of Z-varieties (see chapter 19).

You can read a formula such as χ

ψ

(Frob

q

) ≡ 1 + q − #E(F

q

)

(mod p) in one of two ways. It can tell you what #E(F

q

) (mod p)is,

if you know enough about Frob

q

, or if you know #E(F

q

) (mod p),

it will tell you something about Frob

q

. In practice, it is used in

both ways.

We can also view this formula as a reciprocity law, where the

black box is E, used to produce the number 1 + q − #E(F

q

), which in

turn tells us the trace of Frob

q

acting via the Galois representation

ψ. In fact, more knowledge about elliptic curves sometimes enables

us to ﬁnd the actual conjugacy class of ψ(Frob

q

) after we know the

number 1 + q − #E(F

q

).

As we said already, similar comments apply to viewing the

formula χ

φ

(Frob

q

) ≡ q (mod p) as a reciprocity law for the simple

one-dimensional Galois representation φ. Here the black box was

much simpler: Drop in q and get out q modulo p.

6

For the experts: This does not contradict the fact that an irreducible representation of

a group is determined by its characteristic polynomial, because that fact applies only to

two representations over the same ﬁeld. Here, as we vary p, the ﬁeld F

p

varies. It just

happens that there is one “divine” integer 1 + q − #E(F

q

) whose “avatars” (reductions

mod p) are showing up.

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