Page 178 -

§6. Galois Cohomology Classification of Curves with Given j-Invariant 155

Finally, we make a remark about infinite Galois extensions K/k. The Galois

group Gal(K/k) is the projective limit of all finite quotients Gal(k /k), where k /k
denotes a finite Galois extension contained in K. In this way GalK(K/k) is a com-

pact totally disconnected topological group with Gal(K/k ) forming an open neigh-
borhood basis of the identity. The only G(K/k)-modules that we consider are those

A such that A =∪ k /k A Gal(K/k ) as k goes through finite extensions of k and

where Gal(K/k ) is the kernel Gal(K/k) → Gal(k /k). Then one can prove that

∗
∗
H (Gal(K/k), A) is the inductive limit of H (Gal(k /k), A Gal(K/k ) .Weuse the
notation H (k, A) for H (Gal(k s /k), A), where k s is a separable algebraic closure
∗
∗
of k.
§6. Galois Cohomology Classification of Curves with Given
j j j-Invariant
In3, §2, the invariant j(E) of an elliptic curve E was introduced. In 3(3.2), 3(4.2),
3(5.2) and 3(8.3), we saw that two curves E and E are isomorphic over an alge-

braically closed field k if and only if j(E) = j(E ). Now we take up the question of
the classification of elliptic curves over a perfect field k in the following form. For E

an elliptic curve over a perfect field k, we wish to describe all elliptic curves E /k,up
to isomorphism over k, which become isomorphic to E over the algebraic closure k,

or in other words, all E over k, up to isomorphism over k, with j(E) = j(E ).The
answer given in the next theorem is in terms of the first Galois cohomology group of
Gal(k/k) with values in the automorphism group of the elliptic curve.
(6.1) Theorem. Let E be an elliptic curve over a perfect field k. We have a base
point preserving bijection
 
 Isomorphism classes over k 1

of elliptic curves, E /k with → H Gal k/k , Aut (E)
k
j(E) = j(E ).
 

defined by choosing any isomorphism u : E → E over k and assigning to the class

( u).
of E over k the cohomology class determined by the cocycle a s = u −1 s
( u) is a cocycle by the calculation
Proof. The 1-cocycle a s = u −1 s
" # " #
t
−1 st −1 s s −1 st −1 s −1

a st = u u = u u u u = u u u u
s
= a s · a t for all s, t in Gal(k/k).

If v : E → E is a second isomorphism over k, then h = v −1 u in Aut (E)
k
and from the relation vh = u we have a coboundary relation a s = u −1 s
( u) =
s
h −1 −1 s s −1 b s ( h), where b s is the cocycle b s = v −1 s
( v)( h) = h
( v). Thus the
v
1
cohomology class [a s ] ∈ H (Gal(k/k), Aut (E)) is well defined. Conversely, if
k