Counting rational points on quadric surfaces

Counting rational points on quadric surfaces, Discrete Analysis 2018:15, 29 pp.

A quadric hypersurface of dimension $D$ is the zero set of a quadratic form $Q$ in $D+2$ variables. If $Q(x_1,\dots,x_{D+2})=0$, then $Q(\lambda x_1,\dots,\lambda x_{D+2})=0$ for any $\lambda$, so it is natural to regard a quadric hypersurface of dimension $D$ as a $D$-dimensional variety living in $(D+1)$-dimensional projective space. This paper is concerned with quadratic forms in four variables, which yield 2-dimensional hypersurfaces – that is, surfaces – in the projective space $\mathbb P_3$. As the title suggests, the aim of the authors is to obtain estimates for how many rational points such surfaces contain, or rather, since they contain infinitely many such points, how many there are with at most a given complexity.

A natural way to measure the complexity of a rational point is to multiply it by the smallest scalar that yields a point with integer coordinates and then to take the largest absolute value of any of those coordinates. So if $B$ is a positive integer and $Q$ is a quadratic form in four variables, the authors define $N(B)$ to be the number of integer points $(x_1,x_2,x_3,x_4)$ such that $x_1,\dots,x_4$ do not have a common factor, $Q(x_1,x_2,x_3,x_4)=0$, and $|x_i|\leq B$ for each $i$.

Such estimates had been obtained previously by Browning, but they are strengthened in significant ways in this article, to the point where they are now almost best possible. In particular, the correct dependence on $B$ is obtained. A consequence of the results is that if $Q$ has at least one non-trivial zero, then $N(B)\leq C_Q B^2$ if $\Delta_Q$ is square free and $N(B)\leq C_QB^2\log B$ otherwise. It is conjectured that the dependence of $C_Q$ on $Q$ is unnecessary and that $C_Q$ can be replaced by an absolute constant.

Interesting as these results are, there is more to the paper than simply sharpening some estimates. The authors have gone on to use their results in an important paper that proves a conjecture of Manin for the surface $x_1 y_1^2 + ... + x_4 y_4^2 = 0$ in $\mathbb P_3\times \mathbb P_3$, which is a central conjecture made in 1989 about the distribution of rational points on varieties. See the reference below for details.

T. D. Browning and D. R. Heath-Brown, Density of rational points on a quadric bundle in $\mathbb P_3\times\mathbb P_3$, arxiv:1805.10715