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§7. Bad Reduction and Potentially Good Reduction 121

3
3
2
2
3
3
since j(E) = c / , j(E) − 12 = c / , and 12 = c − c . Conversely, if
4 6 4 6
3
ord p ( j − 12 ) = 0 = ord p ( j), then the equation
12 3 1
2 3
y + xy = x − x −
j − 12 3 j − 12 3
shows that the curve E with j(E) = j can be defined over k.
(7.4) Remark. If E has multiplicative reduction at p, then ord p ( j(E)) < 0since
3
j(E) = c / and ord p (c 4 ) = 0 for multiplicative reduction by (7.2(1)).
4
Now we consider an elliptic curve E over k with j(E) ∈ R, that is, with
ord p ( j(E)) ≥ 0 for all p even at those irreducibles p where E has bad reduction.
Case 1. If k has characteristic unequal to 2, then E can be defined by an equation
2

of the form y = f (x).Ifweextendthe field k to k including the roots of f (x),
2
then E becomes isomorphic to a curve with equation in Legendre form E λ : y =
x(x − 1)(x − λ) where the j-invariant in this case is
2
(λ − λ + 1) 3
8
j(E λ ) = j(λ) = 2
2
λ (λ − 1) 2
by 4(1.4). Given a value j = j(E), we can solve for one of the values of λ such that
j(λ) = j. In general there are six possible λ.Iford p (λ) < 0, then
2
ord p ( j(λ)) = 3 ord p (λ ) − 2 ord p (λ) − 2 ord p (λ) < 0,
and hence ord p ( j) ≥ 0 implies that ord p (λ) ≥ 0. Furthermore, the curve E λ has
good reduction since ¯ λ is not 0 or 1 because it is a solution of j = j(λ) over the
¯
residue class field k(p).
Case 2. If k has characteristic unequal to 3, then E can be transformed into an
2
3
equation of the form E α : y + αxy + y = x where the j-invariant in this case is
3
3
α (α − 24) 3
j(E a ) = j(α) =
3
α − 27
by 4(2.2) and 4(2.3). Given a value j = j(E), we can solve for one of the values of
α such that j(α) = j.Iford p (α) < 0, then
3
3
ord p ( j(α)) = 3 ord p (α) + 3 ord p (α ) − 3 ord p (α )< 0,
and hence ord p ( j) ≥ 0 implies that ord p (α) ≥ 0. Furthermore, the curve E α has
3
¯
good reduction since ¯α is unequal to 27 because it is a solution of j = j(α) over
the residue class field k(p).
The above discussion is a proof of Deuring [1941] characterizing potentially
good reduction. We formulate the concept and theorem for fields K with a discrete
valuation in terms of finite extensions L of K and prolongations w of v to L. Recall
that these extensions w of v always exist.