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§2. The N´ eron Minimal Model 277

(1.3) Corollary. Let E be an elliptic curve over a field K with a complete discrete
valuation. The group E (1) (K) is uniquely divisible by all integers m not divisible by
the characteristic of k.

Proof. For the formal group law E (t 1 , t 2 ), the series [m](x) = mx +· · · has an
inverse function in R[[x]] since m is invertible in R. Hence the map, which is multi-
plication by m on Rπ with group structure a+ E b is an isomorphism. By the theorem
(1.2) it is an isomorphism on E (1) (K). This proves the corollary.

§2. The N´ eron Minimal Model

Now we touch on concepts which go beyond the scope of the methods of this book.
The minimal model E over K is defined by an equation in normal form whose dis-
#
criminant has a certain minimality property. There is another minimal model E , due
to A. N´ eron, which contains additional information about the reduced curve E. This
¯
additional structure plays an important role in analyzing the entire p-adic filtration
on E(K) and in defining the conductor of an elliptic curve in the global theory. The
basic references are N´ eron [1964], Ogg [1967], Serre–Tate [1968], and Tate [1975,
LN 476].

(2.1) Definition. Let E be an elliptic curve over K. The N´ eron minimal model E #
associated with E is a smooth scheme over R together with an isomorphism of curves
#
over K defined θ : E × R K → E such that for any smooth scheme X over Spec(R)
this isomorphism induces an isomorphism
#
Hom R (X, E ) → Hom K (X × R K, E)
#
given by f → θ( f × R K) for f in Hom R (X, E ).
!
The minimal normal form of the cubic equation also defines a scheme E over
!
R which means that there is a map E → Spec(R). Since Spec(R) consists of two
!
!
points η and s, the general fibre E , i.e., the fibre of E over η the open general point,
η
!
is just the given elliptic curve E over K, and the special fibre E , i.e., the fibre E !
s
over s the closed or special point, is just the reduced curve E over k.Wealsouse
¯
!
!
!
!
the notation E = E × R K and E = E × R k for the general and special fibres.
s
η
¯
Observe that the scheme theoretical language allows us to view E over K and E over
k as part of a single algebraic object.
!
(2.2) Remark. The two-dimensional scheme E over Spec(R) is regular if and only
!
if E is nonsingular. By resolving the possible singularities of E , we obtain a new
¯
#
scheme E + over Spec(R).If E denotes the subscheme of smooth points of the
#
map E + → Spec(R), then there is a map E × E + → E + which restricts to
#
#
a group structure on E over Spec(R). This E is the N´ eron model. In order to
#
check the universal property of E , we consider a morphism of the general fibre
of a smooth scheme g : X × R K = X η → E into E over K. The graph G g