Uniformly counting rational points on conics (Q2925465)

Let \(Q(\mathbf{x})\in\mathbb{Z}[x_1,x_2,x_3]\) be a nonsingular quadratic form, isotropic over \(\mathbb{Q}\). The standard counting function for the rational points on \(Q=0\) may be expressed as NEWLINE\[NEWLINEN_Q(B):=\frac{1}{2}\#\{\mathbf{x}\in\mathbb{Z}^3:\, Q(\mathbf{x})=0,\,\text{h.c.f.}(x_1,x_2,x_3)=1,\,||\mathbf{x}||\leq B\},NEWLINE\]NEWLINE where \(||.||\) is a suitable norm. The factor \(\tfrac12\) allows for the fact that \(\mathbf{x}\) and \(-\mathbf{x}\) represent the same rational point.NEWLINENEWLINEOne may show by various methods that NEWLINE\[NEWLINEN_Q(B)\sim c_QBNEWLINE\]NEWLINE as \(B\rightarrow\infty\). Indeed this is a very special case of the work of Franke, Manin and Tschinkel [\textit{J. Franke} et al., Invent. Math. 95, No. 2, 421--435 (1989; Zbl 0674.14012)]. The constant \(c_Q\) involves a product of local factors, in which the factor for the real completion depends on the norm \(||.||\).NEWLINENEWLINEThe goal of the present paper is to establish an asymptotic relation of the above shape, with explicit dependencies on \(Q\) and \(||.||\). Let \(|Q|\) be the maximum modulus of the coefficients of \(Q\), and define NEWLINE\[NEWLINEK_{||.||}=1+ \sup_{\mathbf{x}\not=\mathbf{0}}\frac{||\mathbf{x}||_{\infty}}{||\mathbf{x}||},NEWLINE\]NEWLINE where \(||.||_{\infty}\) is the sup-norm. The paper's result is restricted to norms which are ``isometric'' to either the sup-norm or the Euclidean norm. (The author says that norms \(||.||\) and \(||.||'\) are ``isometric'' when there is a nonsingular real matrix \(A\) such that \(||A\mathbf{x}||=||\mathbf{x}||'\).) For such norms the central theorem says that NEWLINE\[NEWLINEN_Q(B)=c_QB+O(|Q|^5(BK_{||.||})^{1/2}\log(BK_{||.||}))NEWLINE\]NEWLINE for \(B\geq 2\). Very roughly one may think of \(c_Q\) as being typically of size \(|Q|^{-1}\). Thus for a fixed norm \(||.||\) we would get an asymptotic formula as soon as \(B\) is a little larger than \(|Q|^{12}\). Indeed it is pointed out that the exponent 5 in the main result can be improved somewhat. It might be interesting to know what the optimal shape of such a result would be.NEWLINENEWLINEFor the proof one bgins with the special case in which \(Q\) vanishes at \((0,1,0)\). One can then parameterize the rational points, and count them via a calculation involving primitive lattice points of \(\mathbb{Z}^2\) in certain awkward regions. The general case is then reduced to the situation above by showing that, if \(Q\) is isotropic, then it can be converted, via a matrix with suitably bounded entries, into a form which vanishes at \((0,1,0)\).