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§4. Ramification Properties of -Adic Representations of Number Fields 299

ρ w :Gal(k(w)/k(v)) → GL(V ).

The canonical (possibly topological) generator Fr w has image denoted Fr w,ρ in

GL(V ). Two places w and w extending v are conjugate under the Galois group,
and thus Fr w,ρ and Fr w ,ρ are conjugate in GL(V ). Hence they have the same char-

acteristic polynomial which depends only on v and the representation ρ.Weuse the
notation

P v,ρ (T ) = det(1 − Fr w,ρ T )
for this polynomial. Then P v,ρ (T ) is in 1 + TQ [T ] and has degree equal to the
dimension of V .
(4.6) Definition. The polynomials P v,ρ (T ) = det(1−Fr w,ρ T ) are called the charac-
teristic polynomials of the -adic representation ρ of the number field K. Following
Deligne, the characteristic polynomials P v,ρ (T ) can be defined at the ramified primes
v too by
P v,ρ (T ) = det(1 − ρ w (Fr w )T ),

where ρ w :Gal(k(w)/k(v)) → GL(V w ) is the -adic representation constructed as
above on the fixed part V w of V under the inertia subgroup.
We will see in the next few sections that these polynomials are basic for the anal-
ysis of -adic representations. For this we need two types of results, namely that
there are sufficiently many unramified places and that Frobenius elements are suffi-
ciently numberous in the Galois group. This is the case for finite Galois extensions,
i.e., -adic representations with kernel of finite index.

¯
(4.7) Remark. If an -adic representation has a kernel of finite index in Gal(K/K),
then there are only finitely many ramified places. Those places which are ramified in
the finite extension of K correspond to the kernel of the representation by 4.4. For a
general -adic representation, it is not true that all but a finite number of places are
unramified, although the examples Z (1) and T (E) are the ones that we study in
detail will have this property.

The question of which elements in the Galois group are Frobenius elements leads
ˇ
directly to one of the basic results in algebraic number theory, the Cebotarev density
theorem, which we state after some preliminary definitions. Let " K denote the set
of places of the number field K. For each subset X of " K ,wedenoteby N(t, X) the
number of v ∈ X, Nv ≤ t.
(4.8) Definition. A subset X of " K has a density provided the following limit exists:

N(t, X)
d = lim
t→+∞ N(t," K )
and the value d of the limit is called the density of the subset X.