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§2. Primitive Descent Formalism 161

H 1 Q, ϕ E Q = Hom Gal Q/Q , Z/2Z
ϕ

E(Q) → E (Q)
∗ 2
∗
Q / (Q ) .
The homomorphism α is given in 4(5.6) and considered further in 6(3.1) in this
context, and the homomorphism δ is defined in 7(4.2).
2
3
Again let x = m/e and y = n/e be the coordinates of (x, y) on E[a, b] with
rational coefficients in lowest terms. For n = 0wehave x values given by roots of
∗2
2
x(x + ax + b) = 0, and for x = 0 the value α(0, 0) = b(mod Q ), but otherwise

there is a second root x with xx = b.
For n = 0wehave m = 0 and are led to the study of the congruence properties
of m, n, and e. From the equation

2
2 2
3
2
4
2
n = m + am e + bme = m m + ame + be 4
we introduce the greatest common divisor b 1 = (m, b) and factor m = b 1 m 1 and
b 1 b 2 = b where (m 1 , b 2 ) = 1. The equation becomes

2
2
2
2
n = b 1 m 1 b m + ab 1 m 1 e + b 1 b 2 e 4
1 1

2
2
2
= b m 1 b 1 m + am 1 e + b 2 e 4 .
1
1
2
2
Thus b divides n and so b 1 divides n giving a factorization n = b 1 n 1 . Putting this
1
2
relation into the equation and dividing out b , we obtain the relation
1

2
2
2
n = m 1 b 1 m + am 1 e + b 2 e 4 ,
1 1
2
2
4
2
where the two terms m 1 and b 1 m + am 1 e + b 2 e in the factorization of n are
1 1
relatively prime since (b 2 , m 1 ) = 1and (e, m 1 ) = 1.
We can always choose the sign of b 1 so that m 1 is positive, and the above form of
2
the equation implies that m 1 is a square which we write m 1 = M . Then M divides
n 1 from the above equation, and we can write n 1 = MN, with all factorizations
taking place in the integers Z.
(2.3) Assertion. We can summarize the above discussion for a rational point (x, y)
on E[a, b]with y = 0 in terms of a representation as a quotient of integers in the
form
b 1 M 2 b 1 MN
x = and y = .
e 2 e 3
For x and y reduced to lowest terms we have (M, e) = (N, e) = (b 1 , e) = 1. Also,
since (b 2 , m 1 ) = 1, we have (b 2 , M) = (M, N) = 1. The equation for M and N is
2 4 2 2 4
N = b 1 M + aM e + b 2 e .