2.4. The group structure on E(Q) 29 Figure 3. Louis Mordell (1888-1972) and André Weil (1906-1998). the three points P = (1,0), Q = (2,0), and R = (0, −3) are generators of E(Q). This means that any other point on E(Q) can be expressed as a Z-linear combination of P , Q and R. In other words, E(Q) = {a · P + b · Q + c · R : a, b, c ∈ Z}. For instance, (−3, −1) = P + Q, (8, −22) = P + R, (−2, −4) = P − Q, (−1, −4) = Q − R and (3,3) = P − R. The proof of the theorem has three fundamental ingredients: the so-called weak Mordell-Weil theorem E(Q)/mE(Q) is finite for any m ≥ 2 see below) the concept of height functions on abelian groups and the descent theorem, which establishes that an abelian group A with a height function h, such that A/mA is finite (for some m ≥ 2), is finitely generated.

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