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280 14. Elliptic Curves over Local Fields

2
3
2
(a) y + a 1,0 x 1 y 1 + a 3,1 y 1 = πx + a 2,0 x + a 4,1 x 1 + a 6,2 .
1 1 1
The singular point on the fibre, whose local ring is A = R[x, y] m , blows up into the
conic
2
2
(b) y + a 1,0 xy + a 3,1 y = a 2,0 x + a 4,1 x + a 6,2 .
whose discriminant is b 8,2 =−π −2 b 8 for (A2). Further, in (A3) the conic becomes
2
(c) t + a 3,1 t + a 6,2 = 0,
2
2
2
where t = y −ax is given by (y −ax) = y +a 1 xy −a 2 x . The discriminant of the
quadratic (c) is b 6,2 , and thus under the assumptions of (A3) the conic degenerates
3
into two distinct lines. Dividing (a) by x we obtain an equation F(u,v) = π which
1
modulo π factors into L 1 L 2 L 3 . Since the local ring generated by any two factors L 1 ,
L 2 ,or L 3 is regular, equation (a) gives a regular scheme over R with fibre the three
¯
lines L i = 0. Finally, c is the number lines rational over k. One line is rational over
2
k and the others are given by t + a 3,1 t + a 6,2 = 0. Hence the value c = 1or3,but
it is 3 over a large enough k.
Step III. We assume that v(a 6 ) ≥ 2, v(b 8 ) ≥ 3, and v(b 6 ) ≥ 3. With the sin-
¯
gularity at (0, 0) on E this leads to the relations v(a 1 ) ≥ 1, v(a 2 ) ≥ 1, v(a 3 ) ≥ 2,
3
2
v(a 4 ) ≥ 2, and v(a 6 ) ≥ 3. For the cubic polynomial P(t) = t +a 2,1 t +a 4,2 t +a 6,3 ,
the equation of the curve becomes
2
y + a 1,1 x 1 y 1 + a 3,2 y 2 = P(x 1 ).
2
There are three cases depending on the multiplicity of the roots of P(t). If there is
a double root, there are two subcases depending on the multiplicity of the roots of
2
a quadratic polynomials y + a 3,3 y − a 6,4 . If there is a triple root, there are three
2
subcases depending on the multiplicity of the roots of y + a 3,2 y − a 6,4 and the
divisibility of a 4 .
∗
∗
∗
∗
∗
For details and a description of how Cases I ,I ,IV , III , and II come up, see
0 m
Tate [1975, LN 476, pp. 50–52] and for a related analysis Ogg [1967]. We conclude
with Table 1.
ˇ
§3. Galois Criterion of Good Reduction of N´ eron–Ogg–Safareviˇ c
Let K be a local field with valuation ring R and residue class field R → R/Rπ = k.
There is a criterion in terms of the action of Gal(K s /K) on E(K s ) and the subgroups
N E(K s ) for an elliptic curve E over K to have good reduction at k. The idea behind
2
this condition is that the N E(K s ) is isomorphic to (Z/ZN) for N prime to the
characteristic of K, and the following are equivalent:
2
(a) N E(k s ) ns is isomorphic to (Z/ZN) for all N prime to char(k).
¯
(b) E(k s ) ns is complete, i.e., E = E ns .
¯
¯
¯
(c) E has good reduction.