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

By Alexander Gorodnik, François Maucourant, H. Oh. 2006

69 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3e39b748b48c752b34084b79f049a083901cd102

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Akshay Venkatesh
2. P. Michel
3. Alexander Gorodnik
4. Adrian Ioana
5. H. Oh
6. M. Borovoi
7. V. Blomer
8. R. Munshi

In Mathematical Physics

1. Akshay Venkatesh
2. P. Michel
3. Alexander Gorodnik
4. Adrian Ioana
5. H. Oh
6. M. Borovoi
7. Peter Sarnak
8. V. Blomer

Citations in the corpus

• The subconvexity problem for GL2 (2009, background): “16 It should be noted that we could not follow the quantitative arguments of [28] in the case which relies on Clozel’s work; however, their argument certainly furnishes a bound.”
• Effective equidistribution of S-integral points on symmetric varieties (2007, unknown): “Recently similar mixing properties in an adelic setting have been used in the study of rational points of group varieties (see [9], [ 25 ], and [30]).”
• Rational Points on Homogeneous Varieties and Equidistribution of Adelic Periods (2008, background): “1 results solve new cases of Manin’s conjecture [1] on rational points of Fano varieties.”
• Cocycle Superrigidity for Profinite Actions of Property (T) Groups (2008, methodology): “ Ai is of the form r−1 n (Y), for some subset Y ⊂ Xn, for every 1 ≤ i≤ ℓ. This means that vfactors through the map idΓ ×rn, which finishes the proof. ProofofTheorem4.1. We will deduce this result from [GMO08] by following closely the procedurefrom [Lu94, Section 6.3]. Denote by P the set of all prime”