32 2. Elliptic curves

One of the key pieces of evidence in favor of such a conjecture

was offered by Shafarevich and Tate, who proved that there exist

elliptic curves defined over function fields Fp(T ) and with arbitrarily

large ranks (Fp(T ) is a field that shares many similar properties with

Q; see [ShT67]). In any case, the problem of finding elliptic curves

of high rank is particularly interesting because of its arithmetic and

computational complexity.

2.5. The torsion subgroup

In this section we concentrate on the torsion points of an elliptic

curve:

E(Q)torsion = {P ∈ E(Q) : there is n ∈ N such that nP = O}.

Example 2.5.1. The curve En :

y2

=

x3 −n2x

= x(x−n)(x+n) has

three obvious rational points, namely P = (0,0),Q = (−n,0),T =

(n,0), and it is easy to check (see Exercise 2.12.6) that each one of

these points is torsion of order 2, i.e., 2P = 2Q = 2T = O, and

P + Q = T . In fact En(Q)torsion = {O,P,Q,T }

∼

= Z/2Z ⊕ Z/2Z.

Note that the Mordell-Weil theorem implies that E(Q)torsion is

always finite. This fact prompts a natural question: what abelian

groups can appear in this context? The answer was conjectured by

Ogg and proven by Mazur:

Theorem 2.5.2 (Ogg’s conjecture; Mazur, [Maz77], [Maz78]). Let

E/Q be an elliptic curve. Then E(Q)torsion is isomorphic to one of

the following groups:

Z/NZ with 1 ≤ N ≤ 10 or N = 12, or (2.5)

Z/2Z ⊕ Z/2MZ with 1 ≤ M ≤ 4.

Example 2.5.3. For instance, the torsion subgroup of the elliptic

curve with Weierstrass equation y2 + 43xy − 210y = x3 − 210x2 is

isomorphic to Z/12Z and it is generated by the point (0,210). The

elliptic curve

y2

+ 17xy − 120y =

x3

−

60x2

has a torsion subgroup

isomorphic to Z/2Z⊕Z/8Z, generated by the rational points (30, −90)

and (−40,400). See Figure 4 for a complete list of examples with each

possible torsion subgroup.