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
E(Q)torsion = {P E(Q) : there is n N such that nP = O}.
Example 2.5.1. The curve En :
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
+ 17xy 120y =

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.
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