28 2. Elliptic curves Q. Let R be the third point of intersection of L and E. Then R is also a rational point in E(Q). Let L be the line through R and O. We define P + Q to be the third point of intersection of L and E. Notice that any vertical line x = a in the affine plane passes through [0,1,0], because the same line in projective coordinates is given by x = az and [0,1,0] belongs to such line. Thus, if E is given by a model y2 = x3 + Ax + B, and O is chosen to be the point [0,1,0], then L is always a vertical line, so P + Q is always the reflection of R with respect to the x axis. The next step in the study of the structure of E(Q) was conjec- tured by Jules Poincaré in 1908, proved by Louis Mordell in 1922 and generalized by André Weil in his thesis in 1928: Theorem 2.4.3 (Mordell-Weil). E(Q) is a finitely generated abelian group. In other words, there are points P1,...,Pn such that any other point Q in E(Q) can be expressed as a linear combination Q = a1P1 + a2P2 + · · · + anPn for some ai Z. The group E(Q) is usually called the Mordell-Weil group of E, in honor of the two mathematicians who proved the theorem. Example 2.4.4. Consider the elliptic curve E/Q given by the Weier- strass equation y2 + y = x3 7x + 6. The set of rational points E(Q) for this elliptic curve is infinite. For instance, the following points are on the curve: (1,0), (2,0), (0, −3), (−3, −1), (8, −22), (−2, −4), (3, −4), (3,3), (−1, −4), (1, −1), (0,2), (2, −1), (−2,3), (−1,3), 1 4 , 13 8 , 25 9 , 91 27 , 26 9 , 28 27 , 7 9 , 17 27 , . . . . At a first glance, it may seem very difficult to describe all the points on E(Q), including those listed above, in a succinct manner. However, the Mordell-Weil theorem tells us that there must be a finite set of points that generate the whole group. Indeed, it can be proved that
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