On the distribution of integer points on curves of genus zero
Let \(C\subset \mathbb{A}^n\) be a geometrically irreducible affine curve of genus 0 defined over \(\mathbb{Z}\), and let \(C(\mathbb{Z})\) be the set of integral points on \(C\). Let \(\overline{C}\) be the Zariski closure of \(C\) in \(\mathbb{P}^{n}\) and \(C_{\infty} = (\overline{C}\setminus C)(\mathbb{C})\). We denote by \(\Sigma_{\infty}\) the set of discrete valuation rings of the function field of \(C\) at infinity (i.e. the discrete valuation rings which are above a local ring of \(\overline{C}\) at a point of \(C_{\infty}\)). The main result of the paper under review is as follows: Theorem A. The set \(C(\mathbb{Z})\) is infinite if and only if one of the following two conditions is satisfied: (a) \(C_\infty\) consists of one point and \(C(\mathbb{Z})\) contains at least one non-singular point. (b) \(C_\infty\) consists of two points which are conjugate over a real quadratic field and \(C({\mathbf Z})\) contains at least one non-singular point. The part ``only if'' of Theorem A, in the case \(n=2\), is a direct consequence of [\textit{A. Schinzel}, Comment., Pontif. Acad. Sci. 2, No. 20, 1-9 (1969; Zbl 0297.10009)]. Unfortunately, the part ``if'' is not correct, as one can easily see by the following two counterexamples. (1) The equation \(Y^2=(X+1)^2(X^{2}+15)\) defines a rational curve \(C\) with \(C_{\infty}=\{(0\colon 1\colon 0)\}\). Furthermore, the couple \((1,8)\) is a non-singular point on \(C\). Then Theorem A implies that \(C(\mathbb{Z})\) is infinite. If \((x,y)\) is an integer solution of the above equation, then \(x+1\mid y\) and so \(y=(x+1)z\), where \(z\) is an integer. It follows that \(z^{2}- x^{2}=15\), which clearly has finitely many solutions. We deduce that \((X,Y)=(1,\pm 8),(-1,0),(7,\pm 64),(-7,\pm 48)\). Hence part (a) of Theorem A is not correct. (2) Let \(C\) be the rational curve defined by \(f(X,Y)=\) \((X^{2}-2Y^{2})^{2}+XY = 0\). Then \(C_{\infty}=\{(\sqrt{2}\colon 1\colon 0),(-\sqrt{2}\colon 1\colon 0)\}\) and the couple \((1,-1)\) is a non-singular point on \(C\). By Theorem A it follows that \(C(\mathbb{Z})\) is infinite. Let \(x,y\in\mathbb{Z}-\{0\}\) with \(f(x,y)= 0\). Set \(d=\text{gcd}(x,y)\). Then \(x=dx', y= dy'\), where \(x',y'\in\mathbb{Z}\) and \(\text{gcd}(x',y')=1\). Thus \((d({x'}^{2}-2{y'}^{2}))^{2}= |x'y'|\), whence \(|x'|=u^{2}\), \(|y'|=v^{2}\) with \(u,v\in\mathbb{Z}\) and \(u>0, v>0\). It follows that \(d(u^{4}-2v^{4})=uv\) and since \(\text{gcd}(u^{4}-2v^{4},uv)\mid 2\), we get \(u^{4}-2v^{4}\mid 2\). The integer solutions of the equations \(u^{4}-2v^{4} =\pm 1,2\) are well known [cf. \textit{L. J . Mordell}. Diophantine equations, Academic Press, Lond. (1969; Zbl 0188.34503)] and so we finally obtain that the only integer solutions of \(f(X,Y)=0\) are \((X,Y)=(0,0),(1,-1), (-1,1)\). Therefore, part (b) of Theorem A is also not correct. Note that it is possible for a curve \(C\) to satisfy (a) or (b) but at the same time to have more than two discrete valuation rings at infinity (as in the case of the second example), and so Siegel's finiteness theorem implies that \(C(\mathbb{Z})\) is finite. The reviewer believes that the statement of Theorem A can be corrected if we replace the set \(C_{\infty}\) by \(\Sigma_{\infty}\) and make all necessary changes in the proof.
scientific article; zbMATH DE number 1454439