Counting rational points on projective varieties

Abstract

We develop a global version of Heath‐Brown's p‐adic determinant method to study the asymptotic behaviour of the number N(W; B) of rational points of height at most B on certain subvarieties W of Pn defined over Q. The most important application is a proof of the dimension growth conjecture of Heath‐Brown and Serre for all integral projective varieties of degree d ≥ 2 over Q. For projective varieties of degree d ≥ 4, we prove a uniform version N(W; B) = Od,n,ε(Bdim W+ε) of this conjecture. We also use our global determinant method to improve upon previous estimates for quasi‐projective surfaces. If, for example, X́$X\acute{\ }$ is the complement of the lines on a non‐singular surface X ⊂ P3 of degree d, then we show that N(X́;B)=Od(B3/√d(logB)4+B)$N(X\acute{\ };B) = {O}_d( {{B}^{3/\surd d }{{( {\log B} )}}^4 + B} )$ . For surfaces defined by forms a0x0d+a1x1d+a2x2d+a3x3d${a}_0x_0^d + {a}_1x_1^d + {a}_2x_2^d + {a}_3x_3^d$ with non‐zero coefficients, then we use a new geometric result for Fermat surfaces to show that N(X́;B)=Od(B3/√d(logB)4)$N( {X\acute{\ };B} ) = {O}_d({B}^{3/\surd d}{( {\log B} )}^4)$ for B ≥ e.