Paper detail

Liberté et accumulation

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.