Rational points on homogeneous varieties and equidistribution of adelic periods

Alex Gorodnik; Hee Oh; Mikhail Borovoi

doi:10.1007/s00039-011-0113-z

Rational points on homogeneous varieties and equidistribution of adelic periods

Alex Gorodnik, Hee Oh^*, Mikhail Borovoi

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

21 Scopus citations

Abstract

Let U := L\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its normalizer. Assuming that G(K_v) acts transitively on U(K_v) for almost all places v of K, we obtain an asymptotic for the number of rational points U(K) with height bounded by T as T → ∞, and settle new cases of Manin's conjecture for many wonderful varieties. The main ingredient of our approach is the equidistribution of semisimple adelic periods, which is established using the theory of unipotent flows.

Original language	English
Pages (from-to)	319-392
Number of pages	74
Journal	Geometric and Functional Analysis
Volume	21
Issue number	2
DOIs	https://doi.org/10.1007/s00039-011-0113-z
State	Published - Apr 2011

Keywords

Manin conjecture
homogeneous dynamics
rational points
unipotent flows

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title = "Rational points on homogeneous varieties and equidistribution of adelic periods",

abstract = "Let U := L\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its normalizer. Assuming that G(Kv) acts transitively on U(Kv) for almost all places v of K, we obtain an asymptotic for the number of rational points U(K) with height bounded by T as T → ∞, and settle new cases of Manin's conjecture for many wonderful varieties. The main ingredient of our approach is the equidistribution of semisimple adelic periods, which is established using the theory of unipotent flows.",

keywords = "Manin conjecture, homogeneous dynamics, rational points, unipotent flows",

author = "Alex Gorodnik and Hee Oh and Mikhail Borovoi",

note = "Funding Information: Keywords and phrases: Manin conjecture, rational points, homogeneous dynamics, unipotent flows 2010 Mathematics Subject Classification: 11D45, 11E72, 11G35, 28D15, 37A17 A.G. is partially supported by NSF grant 0654413 and RCUK Fellowship. H.O. is partially supported by NSF grant 0629322. M.B. is partially supported by the Hermann Minkowski Center for Geometry and by the ISF grant 807/07",

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doi = "10.1007/s00039-011-0113-z",

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AU - Gorodnik, Alex

AU - Oh, Hee

AU - Borovoi, Mikhail

N1 - Funding Information: Keywords and phrases: Manin conjecture, rational points, homogeneous dynamics, unipotent flows 2010 Mathematics Subject Classification: 11D45, 11E72, 11G35, 28D15, 37A17 A.G. is partially supported by NSF grant 0654413 and RCUK Fellowship. H.O. is partially supported by NSF grant 0629322. M.B. is partially supported by the Hermann Minkowski Center for Geometry and by the ISF grant 807/07

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N2 - Let U := L\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its normalizer. Assuming that G(Kv) acts transitively on U(Kv) for almost all places v of K, we obtain an asymptotic for the number of rational points U(K) with height bounded by T as T → ∞, and settle new cases of Manin's conjecture for many wonderful varieties. The main ingredient of our approach is the equidistribution of semisimple adelic periods, which is established using the theory of unipotent flows.

AB - Let U := L\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its normalizer. Assuming that G(Kv) acts transitively on U(Kv) for almost all places v of K, we obtain an asymptotic for the number of rational points U(K) with height bounded by T as T → ∞, and settle new cases of Manin's conjecture for many wonderful varieties. The main ingredient of our approach is the equidistribution of semisimple adelic periods, which is established using the theory of unipotent flows.

KW - Manin conjecture

KW - homogeneous dynamics

KW - rational points

KW - unipotent flows

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EP - 392

JO - Geometric and Functional Analysis

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IS - 2

ER -

Rational points on homogeneous varieties and equidistribution of adelic periods

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