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.

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