By Dale Husemöller

ISBN-10: 0387954902

ISBN-13: 9780387954905

ISBN-10: 0761517855

ISBN-13: 9780761517856

First variation offered over 2500 copies within the Americas; re-creation includes 3 new chapters and new appendices

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5) Example. This example is related to the Fermat equation w 3 = u 3 + v 3 . It is known that the only rational solutions of this equation are (u, v, w) = (0, 0, 0), (1, 0, 1), (−1, 0, −1), (0, 1, 1), and (0, −1, −1). This cubic equation is not in normal form, but under the transformation x= 3w u+v and y= 9 2 u−v u+v 1 + , 2 we obtain the cubic curve E with equation in normal form y 2 − y = x 3 − 7. §2. Illustrations of the Elliptic Curve Group Law 31 The calculation is an exercise; it is helpful to note that y 2 − y = z 2 − 1/4 when y = z + 1/2.

Hence the set {0, (1, 0), (0, 0), (0, −1), (1, −1)} is a cyclic subgroup of E(Q) of order 5. In fact we will see later that E(Q) is exactly this cyclic group of order 5. The next example differs from the previous one by a single change in sign of the coefﬁcient of x 2 . 3). 4) Example. For the curve E deﬁned by y 2 + y = x 3 + x 2 we have four obvious points (0, 0), (−1, 0) (0, −1) = −(0, 0) and (−1, −1) = −(−1, 0), on the curve. 30 1. Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve The point P = (0, 0) generates an inﬁnite cyclic subgroup of E(Q).

Finally, the third relation between coefﬁcients 3r 2 = a − 2βλ gives the following formula 3 λ4 9 =a−2 λ4 √ 3 3 √ which is impossible for a and λ rational numbers since 3 is irrational. In this theorem and the next the nonexistence of p torsion for p > 3 will be taken up later. 3) Theorem. Let E be the elliptic curve deﬁned by the equation y 2 = x 3 + a, where a is a sixth-power free integer. The torsion subgroup of E(Q) is Z/6Z if a = 1, Z/3Z if a is a square different from 1, or a = −432 = −24 33 , Tors E(Q) = Z/2Z if a is a cube different from 1, 0 if a is not a cube, a square, or −432.