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Fearless Symmetry by Robert Gross, Avner Ash

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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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