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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
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

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