Page 181 -

158 8. Descent and Galois Cohomology

(1.1) Galois Action on the Homogeneous Space. Following 7(3.9) we can de-
s
scribe the action (x, y) of Gal(k s /k) on X(k s ). This can be identified with E(k s )
the locus of a cubic equation in normal form. The action is given by the formula

s
(x, y) = (sx, sy) + a s ,
where (sx, sy) is the action of s ∈ Gal(k s /k) on (x, y) a point of the defining cubic
equation and [a s ] is the cohomology class corresponding to X. Finally, we remark
that the homogeneous space X corresponds to the zero cohomology class if and only
0
if X(k) = H (Gal(k s /k), X(k s )) is nonempty.
(1.2) Notations. Since cohomology for the group G = Gal(k s /k) arises so fre-
i
i
quently, we introduce a special notation, namely H (k,) for H (Gal(k s /k), ).Im-
plicitly, a choice has been made for a separable algebraic closure, and in the case of
a perfect field the separable algebraic closure k s is the algebraic closure.
(1.3) Notations. Let T be any group of order 2 with trivial Gal(k s /k) action, the
1
only action possible. A morphism u is defined on H (k, T ) = Hom(Gal(k s /k), T )
∗ 2
∗
with values in k /(k ) with the property the quadratic extension corresponding to
ker(t) in Gal(k s /k) is generated by the square root of the image u(t). This construc-
tion was used in 7(6.1).
If E(k) has an element of order 2, then we have a morphism T → E(k s ) of
Gal(k s /k)-groups. This induces a morphism
1
1
H (k, T ) = Hom (Gal (k s /k) , T ) → H (k, E (k s )) .
Hence to each quadratic extension k t of k corresponding to a nonzero element t ∈
1
H (k, T ) = Hom(Gal(k s /k), T ) there is a curve P t with P t (k t ) = E(k t ).
(1.4) The Case of the Curve E = E[a, b]. When E(k) has an element of order 2,
we might as well transform the curve by translation so that (0,0) is the point of order
2 and the curve has the form E = E[a, b] which we have studied at length explicitly.
1
The two-element group in question is T ={0(0, 0)}, and for t ∈ H (k, T ) we
denote the corresponding quadratic extension by k t again. The quotient Gal(k t /k)
of Gal(k s /k) has two elements 1 and s where we write frequently x for sx.The
1
image of t in the cohomology group H (k, E(k s )) corresponds to a curve P t where
P t (k t ) = E(k t ) with Galois action given by the following formula where we employ
4(5.4):

b by
(x, y) = (x, y) + (0, 0) = , − .
s P t 2
x x
1
(1.5) Remarks. The element corresponding to P t in H (k, E(k s )) is zero if and
has a fixed point by 7(3.8), or, equivalently, Gal(k t /k) has a fixed point
only if s P t
on P t (k t ) or Gal(k s /k) has a fixed point on P t (k s ). This is also equivalent to the
assertion that P t (k) is nonempty.