Elliptic Curves by Dale Husemöller (auth.)

By Dale Husemöller (auth.)

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Bv + cl. This is the cu~ in the equation of the curve, and, hence, the roots are· v = (X, fJ, y, so that .. = - 2,x, - 2tJ, - 2y. , and UIe tbc relations b = IXtJ + tJy -a .... IX + tJ + y, + yex, and c= - extJy. Thus'tbc cquations forA. ']. - 21l:)2 8y ' A. 1. 2 - ex + tJ + y)2 = = -4exA. 2 - 8y ' A. + 4ßy 4(ex/A. - ßl y')2, where ex/2 = -IX, ß' 2 = - /I, and y'2 = - y. 1. 2 + ß+ y - IX = ±2(ex/A. 1. 1. - IX) = - ß + 2ßl y' - Y or Taking the square root ofboth sides ofthe last equation, we find four solutions forA.

9 LE(s). A refinement of their eonjeeture explains the number lim.... 1 (s The final part of the book is devoted to an elaboration of this eonjeeture. lr §6. Rational Points on Rational Curves. Faltings and the Mordell Conjecture The eases of rational points on curves of degrees 1, 2, and 3 have heen considered, and we are led naturally into the study of elliptic eurves by our simple geometrie approach to diophantine equations. Before going into elliptie eurves, we mention some things about eurves of degree strictly greater than 3.

This question goes back to Diophantus and was taken up by Bachet in 1621. For the case k = 2 we will given an argument based on properties of the ring Z [J-2] in the imaginary quadratic field. We factor (y + J=k)(y - J=k) = x 3 , where x and y must both be odd for the equation to hold mod 4. If p divides x, then p3 divides (y + J-2)(y - J-2). If p divides both factors, then it would divide the sum 2y, and this is impossible from y2 + 2 = x 3 . Thus each factorisaperfectcube,andy + = (a + -2b)3,from which wededuce that J-2 The last equation given b = was asserted.

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