2.10. Homogeneous spaces 59

Corollary 2.9.7 (Weak Mordell-Weil theorem). Let E : y2 = (x −

e1)(x−e2)(x−e3) be an elliptic curve, with ei ∈ Z. Then E(Q)/2E(Q)

is finite.

Proof. By Cor. 2.9.6, E(Q)/2E(Q) injects into ΓΔ ⊂ Γ × Γ × Γ .

Since Γ is finite, E(Q)/2E(Q) is finite as well.

2.10. Homogeneous spaces

In this section we want to make the weak Mordell-Weil theorem ex-

plicit, i.e., we want:

• explicit bounds on the size of E(Q)/2E(Q), and

• a method to find generators of E(Q)/2E(Q) (see Exercise

2.12.25, though).

Before we discuss bounds, we need to understand the structure

of the quotient E(Q)/2E(Q). Remember that, from the Mordell-Weil

theorem (Thm. 2.4.3), E(Q)

∼

= T ⊕

ZRE

where T = E(Q)torsion is a

finite abelian group. Therefore,

E(Q)/2E(Q)

∼

= T/2T ⊕

(Z/2Z)RE

.

In our restricted case, we have assumed all along that E(Q) contains

4 points of 2-torsion, namely O and (ei,0), for i = 1,2,3. And, by

Exercise 2.12.6, E(Q) cannot have more points of order 2. Thus,

T/2T

∼

=

Z/2Z ⊕ Z/2Z (see Exercise 2.12.20).

Hence, the size of E(Q)/2E(Q) is exactly

2RE+2,

under our as-

sumptions. Recall that we defined ν(N) to be the number of distinct

prime divisors of an integer N . We prove our first bound:

Proposition 2.10.1. Let E :

y2

= (x − e1)(x − e2)(x − e3) be an

elliptic curve, with ei ∈ Z. Then the rank of E(Q) is RE ≤ 2ν(ΔE).

Proof. If the quantity ΔE has ν = ν(ΔE) distinct (positive) prime

divisors, then we claim that the set

Γ = {n ∈ Z : 0 = n is square-free and if p | n, then p |

Δ}/(Z×)2

has precisely

2ν(ΔE)+1

elements. Indeed, if ΔE = p11

s

· · · pνν

s

, then

Γ =

{(−1)t0

p11

t

· · · pνν

t

: ti = 0 or 1 for i = 0,...,ν}.