Vojta's Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces

TL;DR

This paper demonstrates that Vojta's Conjecture implies the Batyrev-Manin Conjecture for K3 surfaces by linking rational point distribution to Vojta's predictions in arithmetic geometry.

Contribution

It proves that Vojta's Conjecture implies the Batyrev-Manin Conjecture specifically for K3 surfaces, establishing a significant connection between these conjectures.

Findings

Vojta's Conjecture implies rational points tend to repel each other.

Vojta's Conjecture leads to the Batyrev-Manin distribution predictions.

The paper confirms the Batyrev-Manin Conjecture for K3 surfaces under Vojta's assumptions.

Abstract

Vojta's Conjectures are well known to imply a wide range of results, known and unknown, in arithmetic geometry. In this paper, we add to the list by proving that they imply that rational points tend to repel each other on algebraic varieties with nonnegative Kodaira dimension. We use this to deduce, from Vojta's Conjectures, conjectures of Batyrev-Manin and Manin on the distribution of rational points on algebraic varieties. In particular, we show that Vojta's Main Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces.