2.9. Descent and the weak Mordell-Weil theorem 55

where all the congruences are modulo f(x) = x3 + Ax + B. The

congruences in the previous equation say that a polynomial of degree

1, call it g(x) = x1 − x, is congruent to a polynomial of degree ≤

2, call the last line h(x), modulo a polynomial of degree 3, namely

f(x). Then h(x) − g(x) is a polynomial of degree ≤ 2, divisible by a

polynomial of degree 3. This implies that h(x) − g(x) must be zero

and h(x) = g(x), i.e.,

x1 − x = (2ac +

b2

−

Ac2)x2

+ (2ab −

Bc2

− 2Abc)x +

(a2

− 2bcB).

If we match coeﬃcients, we obtain the following equalities:

2ac +

b2

−

Ac2

= 0, (2.7)

2ab −

Bc2

− 2Abc = −1, (2.8)

a2

− 2bcB = x1. (2.9)

If c = 0, then b = 0 by Eq. (2.7); therefore, p(x) = a +bx+cx2 = a is

a constant function, and so t1 = t2 = t3. By Eq. (2.6), it follows that

e1 = e2 = e3, which is a contradiction with our assumptions. Hence,

c must be non-zero. We multiply Eq. (2.8) by

1

c2

and Eq. (2.7) by

b

c3

to obtain

2ab

c2

− B −

2Ab

c

= −

1

c2

, (2.10)

2ab

c2

+

b3

c3

−

Ab

c

= 0. (2.11)

We subtract Eq. (2.10) from Eq. (2.11) to get:

b

c

3

+ A

b

c

+ B =

1

c

2

.

Hence, the point P = (x0,y0) = (

b

c

,

1

c

) is a rational point on E(Q).

It remains to show that x(2P ) = x(Q). From Eq. (2.11) we deduce

that

a =

Ab

c

−

b3

c3

2b

c2

=

A −

(

b

c

)2

2 ·

1

c

=

A − x0

2

2y0

,