Libert\'e et accumulation

TL;DR

This paper introduces a measure of freedom for rational points on algebraic varieties, aiming to identify points that align with the Batyrev-Manin principle by ensuring their distribution matches conjectural predictions.

Contribution

It proposes a new measure of freedom for rational points to better understand their distribution and to refine the application of the Batyrev-Manin conjecture.

Findings

Points with sufficient freedom are distributed according to a probability measure.

The measure helps distinguish points that conform to the Batyrev-Manin principle.

The approach addresses previous limitations in excluding accumulating domains.

Abstract

The principle of Batyrev and Manin and its variants gives a precise conjectural interpretation for the dominant term for the number of points of bounded height on an algebraic variety for which the opposite of the canonical line bundle is sufficiently positive. As was clearly shown by the counter-example of Batyrev and Tschinkel, the implementation of this principle requires the exclusion of accumulating domains which, are found most of the time by using an induction procedure on the dimension of the variety. However this method does not yield a direct characterisation of the points to be excluded. The aim of this paper is to propose a measure of freedom for rational points so that the points with sufficiently positive freedom are randomly distributed on the variety according to a probability measure on the adelic space introduced by the author in a previous paper. From that point of view the rational points which are sufficiently free ought be the ones which respect the principle of Batyrev and Manin.