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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