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§2. The N´ eron Minimal Model 279

(2.6) Special Fibre of the N´ eron Model. We preserve the notations of (2.3), and
we denote by c the order of = E(K)/E (0) (K) and n the number of connected
#
¯
components of E over k. We divide the classification of the special fibers into three
s
classes. The first is the good and multiplicative reduction cases, the second is three
cases of additive reduction controlled by the quadratic part of the Weierstrass qua-
tion, and the third is four cases of additive reduction controlled by the cubic part of
the Weierstrass equation.
Step I. We start with the coefficients a i , b i ,and c i from a minimal model in
normal form.
(G) The curve E has good reduction which is equivalent to v( ) = 0. In this
#
¯
case E = E , and the special fibre is an elliptic curve E which is the reduction of E.
(M) The curve has multiplicative reduction which is equivalent to v( ) > 0
and v(b 2 ) = 0. We change coordinates such that v(a 3 ), v(a 4 ),and v(a 6 )> 0, i.e.,
so that the singularity of the reduced curve is at (0, 0). Then v(b 2 ) = 0 is equivalent
to v(c 4 ) = 0 since v(b 4 )> 0, and we see that v( j(E)) =−v( ) =−n.In
Table 1 this is the Case I n (or (b n )inN´ eron’s notation), where n is the number of
#

connected components of E over k. Let k be the field of roots over k of the equation
¯
s
2
T +¯a 1 T −¯a 2 . There are two cases for the structure of the connected component
E #0 of the identity.
s
(M1) If k = k, then f = 1, E #0 (k) = k ,and is cyclic of order n = v( ).
∗

s
In this case the curve is isomorphic to the Tate curve E q , where q satisfies v(q) = n.

(M2) If k = k, then f = 1, E #0 (k) = ker(N k /k ), is cyclic of order

s

1if v( ) is odd,
c =
2if v( ) is even.
(A) The curve has additive reduction. These are all the other cases and are
treated in the next two steps.
Step II. In this and the next step assume that v( ) > 0, v(b 2 )> 0, and after
2
change of variable v(a 3 ), v(a 4 ),and v(a 6 )> 0. Recall that b 2 = a + 4a 2 which is
1
2
the discriminant of the quadratic T + a 1 T − a 2 . For local uniformizing parameter
π there is a useful notation a i,m = π −m a i , x m = π −m x,and y m = π −m y for when
the quantities have a suitably high valuation (or order of zero). We distinguish three
cases assuming the above.
(A1) Assume v(a 6 ) = 1. This is the cusp, Case II (or (c1)) in Table 1 with
c = 1, n = 1, and f = v( ) = 2. In this case the Weierstrass model is the N´ eron
model since m = (π, x, y) = (x, y) and A = R[x, y] m is regular.
(A2) Assume v(a 6 ) ≥ 2and v(b 8 ) = 2. Observe that v(a 6 ) ≥ 2 implies that
v(b 6 ), v(b 8 ) ≥ 2. This is Case III (or (c2)) in Table 1 with c = 2, n = 2, f = 2, and
v( ) = 3.
(A3) Assume v(a 6 ) ≥ 2, v(b 8 ) ≥ 3, and v(b 6 ) = 2. This is Case IV (or (c3))
in Table 1 with n = 3, c = 1or3, f = 2, and v( ) = 4.
Under the assumptions of (A2) we can write the equation for the curve in the
form