FROBENIUS 179

to be any element in that conjugacy class. (We brieﬂy

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

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