Approximation diophantienne et distribution locale sur une surface torique II

TL;DR

This paper investigates the local distribution of rational points on a toric surface, proposing an empirical formula aligned with the Batyrev-Manin-Peyre principle, and confirms it through detailed geometric analysis.

Contribution

It introduces an empirical formula for local distribution of rational points on a surface and verifies it for a specific toric surface, including the behavior of special rational curves.

Findings

Existence of a limit measure for rational points distribution.

Asymptotic formula for the critical zoom outside a thin set.

Verification of the empirical formula on a toric surface.

Abstract

We propose an empirical formula for the problem of local distribution of rational points of bounded height. This is a local version of the Batyrev-Manin-Peyre principle. We verify this for a toric surface, on which cuspidal rational curves and nodal rational curves all give the best approximations outside a Zariski closed subset. We prove the existence of a limit measure as well as an asymptotic formula for the critical zoom by removing a thin set.