Institut Henri Poincaré

À la redécouverte des points rationnels

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Integral points of bounded height on toric varieties — reloaded

Antoine Chambert-Loir

We study the analogue for integral points of the Manin-Peyre conjecture about the number and distribution of rational points of bounded height. In view of the successes of the Fourier-theoretic method in the years 1995-2000, the case of equivariant (partial) compactifications of algebraic groups is particularly promising. Ten years ago, after Yuri Tschinkel and I had treated the case of vector groups, we released a preprint that claims to tackle the case of tori. Unfortunately, the proof there is incomplete, and a counterexample of Florian Wilsch demonstrates that the situation is more complicated. The goal of the lecture is to present the Fourier-theoretic method, the kind of output it leads to, and to present the new obstruction that arises, with the hope that it will suffice to describe the asymptotic number and distribution of integral points of bounded height on toric varieties.