2.3. Integral points 23 is an invertible rational map, defined over Q, that sends (0,1) to O, and ψ(0, −1) = (0,0). See Exercise 2.12.4. More generally, any quartic C : V 2 = aU 4 + bU 3 + cU 2 + dU + q2 for some a, b, c, d, q Z is isomorphic over Q to a curve of the form E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6, also defined over Q. The isomorphism is given in [Was08], Theorem 2.17, p. 37. Let E be an elliptic curve over Q given by a Weierstrass equation E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6, ai Q. With a change of variables (x, y) (u−2x, u−3y), we can find the equation of an elliptic curve isomorphic to E given by y2 + (a1u)xy + (a3u3)y = x3 + (a2u2)x2 + (a4u4)x + (a6u6) with coefficients aiui Z for i = 1,2,3,4,6. By the way, this is one of the reasons for the peculiar numbering of the coefficients ai. Example 2.2.6. Let E be given by y2 = x3 + x 2 + 5 3 . We may change variables by x = X 62 and y = Y 63 to obtain a new equation Y 2 = X3 + 648X + 77760 with integral coefficients. 2.3. Integral points In 1929, Siegel proved the following result about integral points E(Z), i.e., about those points on E with integer coordinates: Theorem 2.3.1 (Siegel’s theorem [Sil86], Ch. IX, Thm. 3.1). Let E/Q be an elliptic curve given by y2 = x3 + Ax + B, with A, B Z. Then E has only a finite number of integral points. Siegel’s theorem is a consequence of a well-known theorem of Roth on Diophantine approximation. Unfortunately, Siegel’s theorem is not effective and provides neither a method to find the integral points on E nor a bound on the number of integral points. However, in [Bak90], Alan Baker found an alternative proof that provides an explicit upper bound on the size of the coefficients of an integral solution. More concretely, if x, y Z satisfy y2 = x3 + Ax + B, then max(|x|, |y|) exp((106 · max(|A|, |B|))106).
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