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Fearless Symmetry by Robert Gross, Avner Ash

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146 CHAPTER 12
For example, suppose that we start with the permutation
1 1
2 3
3 4
4 2.
This corresponds to the matrix
V(π) =
1000
0001
0100
0010
in GL(4, R). For example, because π (4) = 2, there is a 1 in the fourth
column and the second row.
We have now defined a morphism V from
{1,2,3,4}
to GL(4, R). (It is
tedious to check that this is a morphism, but it is not difficult.) But
you could view these 0’s and 1’s as being elements of any number
system. So V can be viewed as a linear representation into GL(4, Z)
or GL(4, F
p
)foranyprimep. By the way, this representation is
obviously faithful.
Mod p Linear Representations of the Absolute Galois
Group from Elliptic Curves
We call a linear representation of a group to GL(n, F
p
)a“mod
p representation for short. Here is an example of an interesting
“mod p representation. It is a representation of the absolute Galois
group G, and we can get it from the p-torsion of elliptic curves, but
first we need to know a bit more about the torsion.
THEOREM 12.2: Pick an elliptic curve E and a positive
integer n.LetE[n] be the set of all n-torsion.
5
Then we can
choose two particular elements P and Q inside of E[n]sothat
5
The n-torsion E[n] was defined on page 112. Remember that E[n]hasn
2
elements.
GROUP REPRESENTATIONS 147
every element in E[n] can be written as aP + bQ,wherea and
b run separately over integers from 0 to n 1.
Remember that aP means
a times

P + P +···+P if a is not 0, whereas it
means O, the neutral element of the elliptic curve, if a = 0. Also,
remember that we have already mentioned on page 113 how to
use the n-torsion to get a permutation representation: We pick any
element g of G, and we think about how g permutes the n
2
elements
of E[n].
Now we look at g(P)andg(Q). We know that g(P)andg(Q)
must be elements of E[n], so we can write g(P) = aP + bQ and
g(Q) = cP + dQ,wherea, b, c,andd are all numbers between 0
and n 1. Then our representation applied to g is defined to be the
matrix
r(g) =
ac
bd
.
Now suppose n = p is a prime number and view the matrix
r(g) =
ac
bd
as an element of GL(2, F
p
). You can check that if g
1
and
g
2
are two elements of G,thenr(g
1
g
2
) = r(g
1
)r(g
2
). You have to use
the fact that g(P + mQ) = g(P) + mg(Q) for any integers and m.
6
In any case, r is an honest-to-goodness linear representation.
This representation r has been described in a very abstract way.
We have told you how in theory you can find the matrix r(g). In
actual practice, it can be hard to do. One of the beauties of the
modern theory of elliptic curves is that a tremendous amount of
information can be proved about the representation r, without
having to write it down explicitly in formulas. Instead, mathemati-
cians exploit all the symmetries and relationships implicit in the
definitions of elliptic curves and Galois groups. We will give an
example of one type of information that is known about r,afterwe
define Frobenius elements in G in chapter 16.
6
This is true because all the algebra used to find P + mQ involves only rational
numbers, and so is unchanged when you apply the Galois element g.

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