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

FROBENIUS 179
to be any element in that conjugacy class. (We briey
explained what a conjugacy class is in the previous
chapter.)
In fact, we just lied again: Frob
p
really is a union of
conjugacy classes. To be precise, for any p, we define a set
F (p), which is a union of conjugacy classes inside of the
absolute Galois group G. We let Frob
p
refer to any element
of F (p). Because there is a choice involved here, we have to
be careful when we talk about Frob
p
as if it were a single
element of G.
Now suppose that r is a matrix representation of the absolute
Galois group G, o r a Galois representation, for short. The idea is
that we are never going to have a naked Frob
p
in any formula. We
will always be doing something to Frob
p
first, of which the result
will be the same no matter which element of F (p)wetakeasa
stand-in. We are going to discuss not r(Frob
p
), which might depend
on the particular element of the set F (p) we chose, but χ
r
(Frob
p
).
Because the character of a representation of a group is constant
on all elements of a conjugacy class, the fact that we have a choice
about Frob
p
will not matter when we end up talking only about
χ
r
(Frob
p
).
Good Prime, Bad Prime
Because Frob
p
is not just a single conjugacy class but a whole
bunch of them, it is not true that χ
r
(Frob
p
) is always well-defined
(i.e., independent of choices). W e will always have to make an
assumption about the relationship between r and p that will
eliminate any ambiguity.
There is a concept called “ramification” that we will try to explain
later in this chapter. Again, if you want to skip the explanation,
what you need to know is that every Galois representation r comes
with a set of ramified primes. W e usually call this set S. (Sometimes
mathematicians are not that imaginative.) If p is not in the set S,
then the character value χ
r
(Frob
p
) is well-defined. This means that

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