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)t0pt1 1 · · · pνν t : ti = 0 or 1 for i = 0,...,ν}.

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