Counting rational points on hypersurfaces

Journal article

For any $n \geq 2$, let $F \in \mathbb{Z}[x_1,\ldots,x_n]$ be a form of degree $d\geq 2$, which produces a geometrically irreducible hypersurface in $\mathbb{P}^{n-1}$. This paper is concerned with the number $N(F;B)$ of rational points on $F=0$ which have height at most $B$. For any $\varepsilon>0$ we establish the estimate $ N(F;B)=O(B^{n-2+\varepsilon}),$ whenever either $n\leq 5$ or the hypersurface is not a union of lines. Here the implied constant depends at most upon $d, n$ and $\varepsilon$.

Authors: Heath-Brown, D; Browning, T

• Publication date: 2005-01-01
• DOI: 10.1515/crll.2005.2005.584.83
• ISSN: 0075-4102