P. Salberger - Quantitative aspects of rational points on algebraic varieties (part4)
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Descriptif
Let X be a subvariety of Pn defined over a number field and N(B) be the number of rational points of height at most B on X. There are then general conjectures of Manin on the asymptotic behaviour of N(B) when B goes to infinity. These conjectures can be studied using the Hardy-Littlewood method for non-singular complete intersections of high dimensions and by adelic harmonic analysis for varieties related to algebraic groups. But for most varieties there are no other methods available apart from sieve theory and determinant methods. The latter was first developed for affine plane curves in a paper of Bombieri-Pila and extended to projective varieties by Heath-Brown and myself. The goal of the p-adic version of this method is to show that the set of rational points of height at most B on X in a congruence class satisfy further equations if p is large enough compared to B. One can then proceed by induction with respect to dim(X) to obtain uniform upper estimates for N(B) like a proof of the dimension growth conjecture of Heath-Brown and Serre.
The lectures will focus on the determinant method. We will make essential use of Mumford’s geometric invariant theory too see how stability conditions affect the equidistribution of rational points. This is inspired by Donaldson’s and Tian’s theory of Kähler-Einstein metrics. We will also explain how one can use the theory of volumes of line bundles in Lazarsfeld’s book to improve upon the original bounds of Heath-Brown. If time permits we will also mention a number of striking similarities between the determinant method and methods used in the theory of Diophantine approximation.
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