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298 15. Elliptic Curves over Global Fields and -Adic Representations

(3.4) Corollary. Let u : E → E be an isogeny of elliptic curves over K. For a place

v of K, the curve E has good reduction at v if and only if E has good reduction at
v.
Proof. The isogeny u is a group homomorphism, and it induces an isomorphism

V (E) → V (E ) for all . Hence the result follows immediately from the previous
theorem.
In the previous corollary, if is prime to the degree of u, then u already induces
an isomorphism T (E) → T (E ) as Galois modules.

§4. Ramification Properties of -Adic Representations of
ˇ
Number Fields: Cebotarev’s DensityTheorem
We begin by summarizing the notations of (1.4) and (3.1) relative to a number field
K.

(4.1) Notations. Let K be a number field, and for each place v of K,wehavethe
valuation ring R (v) in K, the local field K v with its valuation ring R v , and the residue
a
class field k(v). The cardinality Nv of k(v) is of the form p .Let L be a Galois exten-
v
sion of K with Galois group Gal(L/K), and let w be a place of L extending a place
v of K. The decomposition subgroup in Gal(L/K) of w is denoted D w , and the in-
ertia subgroup is denoted by I w . The natural epimorphism D w → Gal(k(w)/k(v))
induces an isomorphism D w /I w → Gal(k(w)/k(v)). The group Gal(k(w)/k(v))
is generated by Fr w , where Fr w (a) = a Nv is the Frobenius automorphism. If
k(w)/k(v) is an infinite extension, then Fr w generates Gal(k(w)/k(v)) as a topo-
logical group, i.e., the powers of Fr w are dense.

(4.2) Definition. Let K be a number field, and let ρ :Gal(K/K) → GL(V ) be an
¯
-adic representation. The representation ρ is unramified at a place v of K provided
for each place w of K over K we have ρ(I w ) = 1, i.e., the inertia group I w acts
¯
trivially on V .
¯
(4.3) Examples. For the one-dimensional Galois representation Q (1)(K), the rep-
resentation is unramified at v if and only if v does not divide , i.e., v( ) = 0. For an
elliptic curve E over K, the two-dimensional Galois representation V (E) is unram-
ified at v if and only if E has good reduction at v, see 14(3.3).

¯
(4.4) Remark. Let H = ker(ρ) where ρ :Gal(K/K) → GL(V ) is an -adic repre-
sentation. Let L be the fixed subfield of K corresponding to H. Then ρ is unramified
¯
at v if and only if the extension L/K is unramified at the place v.
(4.5) Remark. If ρ : G → GL(V ) is unramified at v, then ρ|D w : D w → GL(V )
induces a morphism D w /I w → GL(V ) which, composed with the inverse of the
natural isomorphism from (3.1), yields a canonical -adic representation of the field