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156 7. Galois Cohomology and Isomorphism Classification of Elliptic Curves

s
( u) and b s = v
( v), then the coboundary rela-
a s = h −1 b s ( h), where a s = u −1 s −1 s
s s s
s

tion leads to vhu −1 = v h (u −1 ) = (vhu −1 ) and vhu −1 : E → E is an iso-
morphism over k between the two elliptic curves compared with E by u : E → E

and v : E → E both isomorphisms over k.
1
t
For t = [a s ] ∈ H (Gal(k/k), Aut (E)) we define a twisted curve E by the
k
t s s s
requirement that E(k) is the set of all (x, y) in E(k) satisfying ( x, y) = (x, y) =
t
a s (x, y). The difficulty is in showing that E(k) really is the set of k-points on an
elliptic curve over k. We work out in detail that case where the cocycle has values
in the subgroup {+1, −1} contained in Aut (E). Note it is equal to Aut (E) for
k k
3
j(E) = 0, 12 . In this case t is a homomorphism t :Gal(k/k) → Z/2Z. Such
homomorphisms are in one-to-one correspondence with quadratic extensions k t =
√ √ √
k( a) of k with nontrivial automorphism s satisfying s(x + ay) = x − ay.If
2
t
E is given by y = f (x), a cubic polynomial f (x) over k, then E is givenbythe
2
t
equation ay = f (x), and the isomorphism u : E → E is given by

y
u(x, y) = x, √ .
a
In terms of the general construction the relation
√ √ √ √
s
− x + ax , y + ay = a s (x, y) = (x, y) = x − ax , y − ay
becomes in this case x = 0and y = 0since −(x, y) = (x, −y). Finally,

√ t

(x , ay ) is on E(k) if and only if (x , ay ) is on E(k). The other cases, including

characteristic 2, are left to be checked by the reader.

(6.2) Summary. If E and E are two elliptic curves over k with j(E) = j(E ) =
3

0, 12 , then there is a quadratic extension k of k with E and E isomorphic over k .
(6.3) Remark. In 3(6.3) we found that there are five elliptic curves over F 2 up to
isomorphism of which two have j = 1 and three have j = 0. The two curves with
j = 1 are isomorphic over a quadratic extension and Aut(E) is the two element
group. These are the curves E 1 and E 2 , see 3(6, Ex. 1). We leave it to the reader to
explain in terms of Galois cohomology the results in Exercises 1–5 of 3, §6 concern-
ing over which field the pairs of curves over F 2 are isomorphic.