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278 14. Elliptic Curves over Local Fields

of g is a subscheme of the common general fibre X η × E η of all three schemes
# + !
X × E ⊂ X × E → X × E . The key observation is that the closure of G g
→
#
in X × E + is in fact in X × E and is the graph G f of a morphism f : X → E #
corresponding to g in the formulation of the universal property. For details the reader
should see N´ eron [1964].
#
#
(2.3) Remarks. The special fibre E of the N´ eron model E of E is a richer object
s
than the reduction E of E and the group E(k) ns ⊂ E(k). The reduction morphism
¯
¯
¯
#
for the group scheme E over Spec(R) defines an epimorphism for a complete R
#
#
E(K) = E (K) → E (k)
s
with kernel E (1) (K). The original reduction E(k) ns is E #0 (k), where E #0 is the
¯
s
s
#
connected component of the algebraic group E . In particular, we have the isomor-
s
phisms
(1) # (0) #
E(K)/E (K) → E (k) and E(K)/E (K) → E (k)/E(k) ns ,
¯
s s
#
#
where E (k)/E(k) ns is the finite group of connected components of E (k).
¯
s s
Now we come to a basic invariant of an elliptic curve over a local field K.
f
(2.4) Definition. The conductor f (E) = π in R of an elliptic curve E over K is
given by
f = v( ) + 1 − n,
#
where n is the number of connected components of E over k.
¯
s
(2.5) Remark. The ordinal f of the conductor f (E) is a sum

0if E ns is an elliptic curve,


f = f + d, where f = 1if E ns is of multiplicative type, or

2if E ns is of additive type,

and d is zero unless the residue class field k has characteristic 2 or 3, and it is given
in terms of the wild ramification of E(K) as a Galois module, see Ogg [1967], where
the relation for f is considered.
Tate [1975] in LN 476, Antwerp IV gives a detailed version of an algorithm for
describing the special fibre of the N´ eron model. For the construction of the N´ eron
model following (2.2) in terms of the valuations (or orders of zero) v(a i ), v(b i ),
v(c i ),and v( ) of the standard coefficients and the discriminant of the minimal
Weierstrass equation for E, see 5(2.2). In 5(7.2) we considered the three classes of
good, multiplicative, and additive reduction. It is the curves with additive reduction
that divide into seven different families when the special fibre of the N´ eron model is
studied. We now give a version of this classification and a table of the possibilities.