Counting rational points on smooth quartic and quintic surfaces

TL;DR

This paper establishes improved bounds on the number of rational points of bounded height on smooth quartic and quintic surfaces, using advanced techniques involving projective plane sections and uniform Faltings's Theorem.

Contribution

It provides new upper bounds for rational points on degree 4 and 5 surfaces, extending previous results and applying a novel combination of geometric and number-theoretic methods.

Findings

Bound $N_{X'}(B) \,\ll_{K,d,\varepsilon} B^{4/3+\varepsilon}$ for degree 4 and 5 surfaces.

General bound $N_{X'}(B) \,\ll_{K,n,d,\varepsilon} B^{(n+1)/n+\varepsilon}$ for non-uniruled surfaces.

Improves previous unconditional bounds for specific degrees.

Abstract

Let $X\subseteq \mathbb{P}^3$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\prime}}(B)$ be the number of rational points of $X$ of height at most $B$ that do not lie on lines contained in $X$. Assuming a suitable hypothesis on the size of the rank of Abelian varieties, we show that $N_{X^{\prime}}(B)\ll_{K,d,\varepsilon} B^{4/3+\varepsilon}$ for any fixed $\varepsilon>0$. This improves an unconditional bound from Salberger for $d=4$ and $d=5$. The proof, based on an argument of Heath-Brown, consists of cutting $X$ by projective planes and using a uniform version of Faltings's Theorem, due to Dimitrov, Gao, and Habegger, to bound the number of rational points on the plane sections of $X$. More generally, we prove that if $X\subseteq \mathbb{P}^n$ is a non-degenerate non-uniruled smooth projective surface defined over $K$, then $N_{X^{\prime}}(B)\ll_{K,n,d,\varepsilon}B^{\frac{n+1}{n}+\varepsilon}$.