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}.
Previous Page Next Page