By Dale Husemöller (auth.)

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2) Theorem. Let E be the elliptic curve defined by the equation y2 = x 3 + ax, where a is a Jourth-power free integer. The torsion subgroup oJ E(Q) is . if - a is a square, if a = 4, if - a is not a square, or Z/2Z Et> Z/2Z { Tors E(Q) = Z/4Z Z/2Z - 4. In all cases (0, 0) is a point of order 2 since any point of order 2 has the form (x, 0), where x is a root of the cubic equation = x 3 + IIX. In particular, there are three points of order 2 if and only if - a is a square. Next consider the equation 2(x, y) = (0, 0) on E(Q).

We quote some values of g 36 I. Elernentary Properties of the Chord-Tangent Group Law on a Cubic Curve for small a for each of the typeS of curves from their paper which is the first of a two-part discussion leading to the conjectures relating the rank g with the zero of tlie L fundion at s = L Included in Tables 1 and 2 are the cubics y2 = x 3 + 4x and y2 = x 3 - 432 with only a fHlite number of rational points from tbeir relation to the Fermat curve theorem_ This is equivalent to g = O. EXERCISES 1.

0) + c(2. ' foe a = b "'" c = O. §4. Multiplicatieß by 2 on an Elliptic Curve Let E be an elliptiG GUf\fe giveh by an equation y2 = f(x), where f(x) is a cubic polynornial with«J1l1 repeatetl reets. We assume through this sectiOil that the fteld oleoefficients k h .. cl1aracteristic different from 2. The~ is tbe hoiaomorphism E(k) 1. E(k) Jiven by multiplication by 2 in tbe abelian group E(k). We have aIready stlklied the kernel of this homomor,hism; it consisfl er () to:ether with all (x, e), where x is ar~t of fex).