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162 8. Descent and Galois Cohomology

Conversely, any solution of this quartic equation yields a rational point on E[a, b].
This quartic equation came from a down to earth factorization of the coordinates
of a rational point on E[a, b], but in §1 we also derived quartic equations used to de-
scribe principal homogeneous spaces representing elements in a Galois cohomology
group. The Galois cohomology theory is contained in the following exact sequence:

E (Q) 1 1

0 → → H (Q, T ) → ϕ H Q, E Q → 0
ϕE(Q)
2

considered in (2.2), where E = E[a, b]and E = E[−2a, a − 4b]. Elements t in
1
1
H (Q, T ) correspond to quadratic extensions of Q and their image in H (Q, E(Q))
is described by a homogeneous space P t , a curve of genus 1, defined by the quartic
given in (1.7)
4
2
2

N = d M − 2aM + d ,
2
2

where d d = a − 4b. This curve associated with E[−2a, a − 4b] is exactly the
same curve as the quartic above
2 2
4
2
N = b 1 M + aM e + b 2 e 4
1

associated with E[a, b]. The element t in H (Q, T ) comes from a point in E (Q)/
ϕE(Q) if and only if this quartic in (1.7) has a rational point, i.e., the principal
homogeneous space is trivial.
The methods described above do allow one to calculate the rank of certain elliptic
curves over Q.
3
2
(2.4) Example. Consider the curve E = E[0, p] with equation y = x − px for p
2
3
a prime number. Then E is E[0, 4p] with equation y = x +4px. For the first curve

the divisors of b =−p are b =±1, ±p, and the corresponding quartic equations
are
2 4 4 2 4 4 2 4 4 2 4 4
N = M − pe , N =−M + pe , N = pM − e , N =−pM + e .
For p ≡ 3(mod4), −1 is not a square mod p so that the second and third equations
have no solution mod p, and, hence, no solution in the integers. This means that for
p ≡ 3(mod4)

∗ 2
im(α on E) ={1, −p} mod Q .
For the second curve the divisors of b = 4p are b =±1, ±2, ±4, ±p, ±2p, ±4p,

and the corresponding quartic equations are
4
4
4
4
2
2
2
4
4
N = M + 4pe , N = 2M + 2pe , N = 4M + pe ,
2
where we excluded the cases where both factors of 4p are negative since N is
positive. For p ≡±3 (mod 8), 2 is not a square mod p so that the second equation