Large integral points on elliptic curves.

*(English)*Zbl 0611.10008In this interesting note the author studies two aspects of the topic suggested by his title.

(i) Given an elliptic curve \(E\) over \(\mathbb Q\), how can one efficiently search for integral points ?

(ii) How can one construct elliptic curves (over \(\mathbb Q\)) which possess a large integral point ?

Let \(E\) be given in Weierstrass form \(y^ 2=x^ 3+ax+b\) with \(a,b\in\mathbb Z\). The author gives three answers to question (i), each of which requires \(O(\log\log H)\) steps to search for integral solutions with \(| x|, | y| \leq H\). All three depend ultimately on using continued fraction or related algorithms to find approximations of the form \(\alpha r-\beta s\approx 0\) or \(\alpha r-\beta s\approx \gamma\) with \(r,s\in\mathbb Z\), where \(\alpha,\beta\), and \(\gamma\) are fixed real numbers. The first method requires that \(x^ 3+ax+b\) factors completely over \(\mathbb Q\), and uses this factorization in a well-known fashion to reduce the problem to certain Pellian equations, and thence to the above continued fraction equations. The second method requires knowing a basis for the group of rational points \(E(\mathbb Q)\), and uses a combination of canonical heights and Pellian equations. It is superceeded by the third method, which also requires knowledge of \(E(\mathbb Q)\). This method uses the elliptic logarithm \(\phi: E(\mathbb R)^ 0\to\mathbb R/\mathbb Z\), given by the usual incomplete elliptic integral, to reduce directly to a continued fraction equation.

For example, suppose \(E(\mathbb Q)\) has rank one, generated by \(P\) and torsion. Then the author shows that there is an effectively computable constant \(c\) so that any integral point has the form \(rP+T\) for some \(T\in E(\mathbb Q)_{\text{tor}\, s}\) and integers \(r\) and \(s\) satisfying \[ r\cdot 2\phi (P)-s=O(e^{-cr^ 2/2}). \] This last equation can be rapidly solved to high accuracy once one knows \(\phi(P)\). The author gives a new, extremely clever way of computing \(\phi(P)\) which takes \(O(\log B)\) steps to produce \(B\) digits of accuracy. Although older algorithms work in \(O(\log\log B)\) steps the author’s method has the advantage that it only requires elementary arithmetic operations. As an example, the author studies integral points on the curve \(y^ 2=x^ 3-30x+133\). Using 50 digits of \(\phi(P)\), he finds all integral points under about \(10^{10^{50}}\), including the “large” point \((5143326,\;11664498677)\). Finally he notes that the bounds obtainable from his method are barely in the range of the best known upper bounds arising from linear forms in logarithms. Thus if \(E(\mathbb Q)\) has small coefficients and small rank, then one might actually be able to prove that the set of integral points produced by the search is complete.

The author approaches his second topic, the production of curves with large integral points, in a more ad hoc fashion. He starts by defining a measure of impressiveness of an integral point \(P=(x,y)\) on \(E\) by the formula \[ \rho =\log | x| /\log (\max \{| a|^{1/2}, | b|^{1/3}\}). \] On probabilistic grounds, he suggests that for any \(\varepsilon >0\) there should be only finitely many examples having \(\rho >10+\varepsilon\). (This bound was also proposed by S. Lang and H. Stark.) A computer search found a number of examples with \(\rho\) in the 9 to 12 range; and the author produces two 1-parameter families with \(\rho\to 9\). Recently, P. Vojta [Diophantine approximations and value distribution theory (Lect. Notes Math. 1239), Ch. 5, Sect. 5 (1987; Zbl 0609.14011)] has shown that his general conjecture on Diophantine approximations implies the author’s conjecture except possibly for a finite number of 1-parameter families. N. Elkies (letter to D. Zagier, 26/2/87) has found such an exceptional family (of the form \(y^ 2=x^ 3+132x+b(t))\) satisfying \(\rho\to 12\).

(i) Given an elliptic curve \(E\) over \(\mathbb Q\), how can one efficiently search for integral points ?

(ii) How can one construct elliptic curves (over \(\mathbb Q\)) which possess a large integral point ?

Let \(E\) be given in Weierstrass form \(y^ 2=x^ 3+ax+b\) with \(a,b\in\mathbb Z\). The author gives three answers to question (i), each of which requires \(O(\log\log H)\) steps to search for integral solutions with \(| x|, | y| \leq H\). All three depend ultimately on using continued fraction or related algorithms to find approximations of the form \(\alpha r-\beta s\approx 0\) or \(\alpha r-\beta s\approx \gamma\) with \(r,s\in\mathbb Z\), where \(\alpha,\beta\), and \(\gamma\) are fixed real numbers. The first method requires that \(x^ 3+ax+b\) factors completely over \(\mathbb Q\), and uses this factorization in a well-known fashion to reduce the problem to certain Pellian equations, and thence to the above continued fraction equations. The second method requires knowing a basis for the group of rational points \(E(\mathbb Q)\), and uses a combination of canonical heights and Pellian equations. It is superceeded by the third method, which also requires knowledge of \(E(\mathbb Q)\). This method uses the elliptic logarithm \(\phi: E(\mathbb R)^ 0\to\mathbb R/\mathbb Z\), given by the usual incomplete elliptic integral, to reduce directly to a continued fraction equation.

For example, suppose \(E(\mathbb Q)\) has rank one, generated by \(P\) and torsion. Then the author shows that there is an effectively computable constant \(c\) so that any integral point has the form \(rP+T\) for some \(T\in E(\mathbb Q)_{\text{tor}\, s}\) and integers \(r\) and \(s\) satisfying \[ r\cdot 2\phi (P)-s=O(e^{-cr^ 2/2}). \] This last equation can be rapidly solved to high accuracy once one knows \(\phi(P)\). The author gives a new, extremely clever way of computing \(\phi(P)\) which takes \(O(\log B)\) steps to produce \(B\) digits of accuracy. Although older algorithms work in \(O(\log\log B)\) steps the author’s method has the advantage that it only requires elementary arithmetic operations. As an example, the author studies integral points on the curve \(y^ 2=x^ 3-30x+133\). Using 50 digits of \(\phi(P)\), he finds all integral points under about \(10^{10^{50}}\), including the “large” point \((5143326,\;11664498677)\). Finally he notes that the bounds obtainable from his method are barely in the range of the best known upper bounds arising from linear forms in logarithms. Thus if \(E(\mathbb Q)\) has small coefficients and small rank, then one might actually be able to prove that the set of integral points produced by the search is complete.

The author approaches his second topic, the production of curves with large integral points, in a more ad hoc fashion. He starts by defining a measure of impressiveness of an integral point \(P=(x,y)\) on \(E\) by the formula \[ \rho =\log | x| /\log (\max \{| a|^{1/2}, | b|^{1/3}\}). \] On probabilistic grounds, he suggests that for any \(\varepsilon >0\) there should be only finitely many examples having \(\rho >10+\varepsilon\). (This bound was also proposed by S. Lang and H. Stark.) A computer search found a number of examples with \(\rho\) in the 9 to 12 range; and the author produces two 1-parameter families with \(\rho\to 9\). Recently, P. Vojta [Diophantine approximations and value distribution theory (Lect. Notes Math. 1239), Ch. 5, Sect. 5 (1987; Zbl 0609.14011)] has shown that his general conjecture on Diophantine approximations implies the author’s conjecture except possibly for a finite number of 1-parameter families. N. Elkies (letter to D. Zagier, 26/2/87) has found such an exceptional family (of the form \(y^ 2=x^ 3+132x+b(t))\) satisfying \(\rho\to 12\).

Reviewer: Joseph H. Silverman (Providence)

##### MSC:

11D25 | Cubic and quartic Diophantine equations |

14H52 | Elliptic curves |

14G25 | Global ground fields in algebraic geometry |