O'Reilly logo

Fearless Symmetry by Robert Gross, Avner Ash

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

188 CHAPTER 16
For instance, consider f (x) = x
2
11. The discriminant of f is 44.
So consider some prime p different from 2 or 11. We use Theorem
16.1 to compute Frob
p
(
11). (Remember that
11 is by definition
the positive square root, which is approximately 3.3166247904.)
We know Frob
p
on the set of roots {
11,
11} is either the
identity permutation (the neutral element in
{
11,
11}
)orthe
permutation that switches the two roots. In the first case, Frob
p
has two cycles each of length 1, and in the second case, it has only
one cycle of length 2.
Now f (x) factors in F
p
into two simpler factors if and only if 11
is a square modulo p.
5
So, by Theorem 16.1, the cycle type of Frob
p
is 1 + 1 if 11 is a square modulo p, and is 2 otherwise. In the first
case, Frob
p
(
11) =
11 and in the second Frob
p
(
11) =−
11. For
example, Frob
89
(
11) =
11 because x
2
11 factors into two linear
factors in F
89
.
APPENDIX
The Official Definition of the Bad Primes
for a Galois Representation
Suppose we have a Galois representation r. Remember that this
means r is a morphism of G
Q
, the absolute Galois group of Q (which
we are abbreviating G) to some other group H,eitheragroupof
permutations or a group of matrices.
Any morphism f from a group G to a group H has what is called
a kernel. This is just the set of elements x in G with the property
that f (x) is the neutral element e of H. We call the kernel of r by
the name K
r
. Because finding the kernel is the same as solving the
equation f (x) = e, you can imagine that this is an important set to
study. For example, r is a faithful representation precisely when K
r
contains only the neutral element e.
So K
r
is a subset of G. It actually is a group all by itself, but that
is not so important right now.
5
That is, if and only if 11 = a
2
for some a in F
p
.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required