Manin's and Peyre's conjectures on rational points and adelic mixing

Creators: Gorodnik, Alexander; Maucourant, François; Oh, Hee

Abstract

Let X be the wonderful compactification of a connected adjoint semisimple group G defined over a number field K. We prove Manin's conjecture on the asymptotic (as T → ∞) of the number of K-rational points of X of height less than T, and give an explicit construction of a measure on X(A), generalizing Peyre's measure, which describes the asymptotic distribution of the rational points G(K) on X(A). Our approach is based on the mixing property of L^2(G(K)\G(A)) which we obtain with a rate of convergence.

Additional Information

© 2008 Elsevier. The first and the third authors are partially supported by DMS-0400631, and DMS-0333397 and DMS-0629322 respectively. The second author would like to thank Caltech for the hospitality during his visit where the work was first conceived. We would like to thank Wee Teck Gan, Emmanuel Peyre and Yehuda Shalom for helpful conversations. We thank Ramin Takloo-Bighash for the useful comments on our preliminary version of this paper. We are also deeply grateful to the referee for many detailed comments on the submitted version.

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Manin's and Peyre's conjectures on rational points and adelic mixing

Abstract

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