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ˇ
§3. Galois Criterion of Good Reduction of N´ eron–Ogg–Safareviˇ c 281
For the multiplicative group we have N G m (k s ) = µ N (1)(k s ) and for the additive
group we have N G a (k s ) = 0. Thus it is exactly the case of good reduction where the
¯
reduction morphism E(K s ) → E(k s ) restricts to an isomorphism N E → N E,and it
¯
is the case of bad reduction when it has a nontrivial kernel. Unfortunately the above
outline of ideas does not work for the Weierstrass model E over R, but we must use
#
the N´ eron model E of E which has additional naturality properties.
The Galois group Gal(K s /K) → Gal(k s /k) maps surjectively and the kernel is
called the inertia subgroup I of Gal(K s /K).
(3.1) Definition. Aset S on which Gal(K s /K) acts is called unramified provided
the inertia subgroup I acts as the identity on the set S.
In other words, in the unramified case, the action of Gal(K s /K) factors through
an action of the Galois group Gal(k s /k) of the residue class field. This definition can
apply to S = N E(K s ) or T (E) associated withan elliptic curve E over K.

ˇ
(3.2) Theorem (Criterion of N´ eron–Ogg–Safareviˇ c). Let K be a local field with
perfect residue class field k of characteristic p. Then the following assertions are
equivalent for an elliptic curve E over K:
(1) The elliptic curve E has good reduction.
(2) The N-division points N E(K s ) are unramified for all N prime to the character-
istic p of k.

(2) The N-division points N E(K s ) are unramified for infinitely many N prime to the
characteristic p of k.
(3) The Tate module T (E) is unramified for some prime unequal p.
Now we sketch the proof given in Serre–Tate [1968]; see also Ogg [1967]. The
line of argument that we give for elliptic curves works equally well for higher-
dimensional abelian varieties.

Proof. Since T (E) is unramified if and only if each E(K s ) is unramified, it is clear
n

that (2) implies (3) and (3) implies (2) . In order to see that (1) implies (2) and (2)
implies (1), we introduce the notation L for the fixed field of the inertial subgroup I
¯
of Gal(K s /K) and R L for the ring of integers in L.Then k is the residue class field
#
of both R L and R s of K s .For theN´ eron model E we have two isomorphisms and a
reduction morphism r in the sequence
I # r #
¯
E(K s ) → E(L) → E (R L ) → E (k).
s
The basic properties of r are derived from the facts that R L is Henselian, i.e.,
#
Hensel’s lemma applies to it, and that E is smooth. The morphism r is surjective
and Gal(K s /K) → Gal(k/k) equivariant, and the kernel ker(r) is uniquely divisible
¯
by any N prime to p.
#
If E has good reduction, i.e., (1) is satisfied, then E is an elliptic curve so that
s
2
I
# ¯
N E (k) is isomorphic to (Z/NZ) . Thus the same is true for N E(K s ) so that I acts
s
trivially on N E(K s ) for all N prime to p. Thus (1) implies (2).