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§3. Galois Representations and the N´ eron–Ogg–Safareviˇ c Criterion 297
(3.1) Notations. Let L be a Galois extension 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 D w
of Gal(L/K) consists of all s in Gal(L/K) with ws = w. By reduction modulo the
maximal ideal, we have an epimorphism of groups
D w → Gal(k(w)/k(v))

with kernel the inertia subgroup I w of the place w. There is also an isomorphism
given by extension to the completed fields

D w → Gal(L w /K v )
An element s of the decomposition group D w belongs to the inertia subgroup I w of
D w if and only if w(s(a) − a)> 0 for all a in R (w) , or equivalently, in R w . When
the maximal ideal of R (w) or R w is generated by an element π, then an element s in
D w is also in I w if and only if w(s(π) − π) > 0.
Let E be an elliptic curve over K.ThenGal(K/K) acts on E(K),on N E(K),
¯
¯
¯
and on T (E) for all rational primes . Further, for any place w of K extending a
¯
place v of K, the decomposition group D w acts on E(K),on N E(K), and on T (E).
¯
¯
Under the isomorphism D w → Gal(K w /K v ), the following natural homomorphisms
¯
are equivariant morphisms of groups
¯
E(K) → E(K w ), N E(K) → N E(K w ), and T (E/K) → T (E/K v ),
¯
¯
¯
where the second two are isomorphisms. With these remarks, we can translate the
ˇ
local N´ eron–Ogg–Safareviˇ c criterion 14(3.2) into a global assertion using the fol-
lowing definition.
(3.2) Definition. Aset S on which Gal(L/K)) acts is called unramified at a place
v of K provided the inertia subgroups I w act as the identity on the set S for all w|v,
that is, places w of L dividing v.
(3.3) Theorem. Let K be an algebraic number field, and let v be a place of K with
residue class field of characteristic p v . Then the following assertions are equivalent
for an elliptic curve E over K:
(1) The elliptic curve E has good reduction at v.
(2) The N-division points N E(K) are unramified at v for all N prime to p v .
¯

(2) The N-division points N E(K) are unramified at v for infinitely many N prime
to p v .
(3) The Tate module T (E) is unramified at v for some prime unequal to p v .
(3) The Tate module T (E) is unramified at v for all primes unequal to p v .

Q could be used in conditions (3)
In addition, the vector space V (E) = T (E)⊗ Z

and (3) in place of T (E).
This theorem suggest that a systematic study of Q vector spaces with a Gal(K/K)
¯
action should be undertaken. This is the subject of the next sections. It was first done
by Taniyama [1957] and extended considerably by Serre.