×

zbMATH — the first resource for mathematics

Hardy-Littlewood varieties and semisimple groups. (English) Zbl 0917.11025
Let \(X(A)= X(\mathbb{R})\times X(A_f)\) be the set of adelic points of an algebraic variety \(X\) over \(\mathbb{Q}\) (here \(A_f\) is the \(\mathbb{Q}\)-algebra of finite adeles). Let \(B_f\) be a compact subset of \(X(A_f)\), let \(B_0(T)=\{x\mid x\in B_0\), \(| x|\leq T\}\) be the ball of radius \(T\) in a connected component \(B_0\) of \(X(\mathbb{R})\), and let \(B(T)= B_0(T)\times B_f\). The authors are interested in the asymptotic behaviour of the counting function \[ {\mathcal N}_X (T,B)= \text{card} \bigl(X(\mathbb{Q}) \cap B(T) \bigr), \] as \(T\to\infty\). The variety \(X\) is said to be strongly (respectively relatively) Hardy-Littlewood if \({\mathcal N}_X(T,B)\sim\mu(B(T))\) (respectively \({\mathcal N}_X(T,B) \sim\int_{B(T)} \delta (x) d\mu(x)\) for a suitable non-negative function \(\delta:X(A) \to \mathbb{R})\), where \(\mu\) is the Tamagawa measure on \(X(A)\). The authors write: “We prove that certain affine homogeneous spaces are strongly Hardy-Littlewood, \(\dots\) that many homogeneous spaces are relatively Hardy-Littlewood, but not strongly Hardy-Littlewood. This yields a new class of varieties, for which the asymptotic density of integer points can be computed in terms of a product of local densities”.
Reviewer: B.Z.Moroz (Bonn)

MSC:
11G35 Varieties over global fields
14M17 Homogeneous spaces and generalizations
11N45 Asymptotic results on counting functions for algebraic and topological structures
11P55 Applications of the Hardy-Littlewood method
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [Bi] B.J. Birch: Forms in many variables. Proc. Roy. Soc. Ser. A265 (1962) 245-263 · Zbl 0103.03102 · doi:10.1098/rspa.1962.0007
[2] [B-HC] A. Borel, Harish-Chandra: Arithmetic subgroups of algebraic groups. Ann. Math.75 (1962) 485-535 · Zbl 0107.14804 · doi:10.2307/1970210
[3] [Brl] A. Borel: Some finiteness properties of adele groups over number fields. Publ. Math.16 (1963) 101-126 · Zbl 0135.08902
[4] [Bo1] M.V. Borovoi: On strong approximation for homogeneous spaces. Doklady Akad. Nauk BSSR33 (1989) 293-296 (Russian)
[5] [Bo2] M.V. Borovoi: The algebraic fundamental group and abelian Galois cohomology of reductive algebraic groups, Preprint, MPI/89-90, Bonn
[6] [Bo3] M.V. Borovoi: On weak approximation in homogeneous spaces of algebraic groups. Soviet Math. Doklady42 (1991) 247-251
[7] [Bo4] M.V. Borovoi: The Hasse principle for homogeneous spaces. J. Reine Angew. Math.426 (1992) 179-192 · Zbl 0739.14030 · doi:10.1515/crll.1992.426.179
[8] [Bo5] M.V. Borovoi: Abelianization of the second nonabelian Galois cohomology. Duke Math. J.72 (1993) 217-239 · Zbl 0849.12011 · doi:10.1215/S0012-7094-93-07209-2
[9] [Ca] J.W.S. Cassels: Rational Quadratic Forms. London: Academic Press, 1978
[10] [CS] J.H. Conway, N.J.A. Sloane: Sphere Packings, Lattices and Groups, 2nd edn. New York: Springer 1993 · Zbl 0785.11036
[11] [Da] H. Davenport: Analytic Methods for Diophantine Equations and Diophantine Inequalities. Ann Arbor, Michigan: Ann Arbor Publishers, 1962
[12] [Di] J. Dixmier: Quelques aspects de la théorie des invariants. Gaz. Mathematiciens43 (1990) 39-64
[13] [DRS] W. Duke, Z. Rudnick, P. Sarnak: Density of integer points on affine homogeneous varieties. Duke Math. J.71 (1993) 143-179 · Zbl 0798.11024 · doi:10.1215/S0012-7094-93-07107-4
[14] [EM] A. Eskin, C. McMullen: Mixing, counting, and equidistribution in Lie groups. Duke Math. J.71 (1993) 181-209 · Zbl 0798.11025 · doi:10.1215/S0012-7094-93-07108-6
[15] [EMS] A. Eskin, S. Mozes, N. Shah: Unipotent flows and counting lattice points on homogeneous spaces, Preprint · Zbl 0852.11054
[16] [ERS] A. Eskin, Z. Rudnick, P. Sarnak: A proof of Siegel’s weight formula. Duke Math. J., Int. Math. Res. Notices5 (1991) 65-69 · Zbl 0743.11023 · doi:10.1155/S1073792891000090
[17] [Esk] A. Eskin: Ph. D. Thesis, Princeton University 1993
[18] [Es] T. Estermann: A new application of the Hardy-Littlewood-Kloosterman method. Proc. London Math. Soc.12 (1962) 425-444 · Zbl 0105.03606 · doi:10.1112/plms/s3-12.1.425
[19] [FMT] J. Franke, Yu. I. Manin, Yu. Tschinkel: Rational points of bounded height on Fano varieties. Invent. Math.95 (1989) 421-435 · Zbl 0674.14012 · doi:10.1007/BF01393904
[20] [Ha] G. Harder: Über die Galoiskohomologie halbeinfacher Matrizengruppen, I, Math. Z. 90 (1965) 404-428; II, Math. Z.92 (1966) 396-415 · Zbl 0152.00903 · doi:10.1007/BF01112362
[21] [HB] D.R. Heath-Brown: The density of zeros of forms for which weak approximation fails. Math. Comp.59 (1992) 613-623 · Zbl 0778.11017 · doi:10.1090/S0025-5718-1992-1146835-5
[22] [Ig] J.-I. Igusa: Lectures on Forms of Higher Degree. Bombay: Tata Institute of Fundamental Research 1978
[23] [Kn1] M. Kneser: Galoiskohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern, I, Math. Z.88 (1965) 40-47; II, Math. Z.89 (1965) 250-272 · Zbl 0143.04702 · doi:10.1007/BF01112691
[24] [Kn2] M. Kneser: Starke Approximation in algebraischen Gruppen, I. J. Reine Angew. Math.218 (1965) 190-203 · Zbl 0143.04701 · doi:10.1515/crll.1965.218.190
[25] [Ko1] R.E. Kottwitz: Stable trace formula: cuspidal tempered terms. Duke Math. J.51 (1984) 611-650 · Zbl 0576.22020 · doi:10.1215/S0012-7094-84-05129-9
[26] [Ko2] R.E. Kottwitz: Stable trace formula: elliptic singular terms. Math. Ann.275 (1986) 365-399 · Zbl 0591.10020 · doi:10.1007/BF01458611
[27] [Ko3] R.E. Kottwitz: Tamagawa numbers. Ann. Math.127 (1988) 629-646 · Zbl 0678.22012 · doi:10.2307/2007007
[28] [La] S. Lang: Algebraic groups over finite fields. Am. J. Math.78 (1956) 555-563 · Zbl 0073.37901 · doi:10.2307/2372673
[29] [Mi] J.S. Milne: The points of Shimura varieties modulo a prime of good reduction. The Zeta Functions of Picard Modular Surfaces, Montreal 1992, pp. 151-253
[30] [Min] Kh.P. Minchev: Strong approximation for varieties over an algebraic number field. Doklady Akad. Nauk BSSR33 (1989) 5-8 (Russian)
[31] [O] T. Ono: On the relative theory of Tamagawa numbers. Ann. Math.82 (1965) 88-111 · Zbl 0135.08804 · doi:10.2307/1970563
[32] [Pa] S.J. Patterson: The Hardy-Littlewood method and diophantine analysis in the light of Igusa’s work. Math. Gött. Schriftenr. Geom. Anal.11 (1985) 1-45
[33] [Pl] V.P. Platonov: The problem of strong approximation and the Kneser-Tits conjecture for algebraic group. Math. USSR Izv.3 (1969) 1139-1147; Supplement to the paper ?The problem of strong approximation...?. Math. USSR Izv.4 (1970) 784-786 · Zbl 0217.36301 · doi:10.1070/IM1969v003n06ABEH000838
[34] [PR] V.P. Platonov, A.S. Rapinchuk: Algebraic Groups and Number Theory Moscow: Nauka 1991 (Russian; an English translation to be published by Academic Press)
[35] [Ro] M. Rosenlicht: Toroidal algebraic groups. Proc. A.M.S.12 (1961) 984-988 · Zbl 0107.14703 · doi:10.1090/S0002-9939-1961-0133328-9
[36] [Sa] J.-J. Sansuc: Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math.327 (1981) 12-80 · Zbl 0468.14007 · doi:10.1515/crll.1981.327.12
[37] [Sch] W.M. Schmidt: The density of integer points on homogeneous varieties. Acta. Math.154 (1985) 243-296 · Zbl 0561.10010 · doi:10.1007/BF02392473
[38] [Se1] J.-P. Serre: Cohomologie galoisienne. (Lect. Notes Math. vol. 5) Berlin Heidelberg New York: Springer 1965
[39] [Se2] J.-P. Serre: Resumés des cours de 1981-1982, 1982-1983 (Euvres, pp. 649-657, 669-674) Berlin Heidelberg New York: Springer 1986
[40] [Sie1] C.L. Siegel: Über die analytische Theorie der quadratischen Formen II. Ann. Math.37 (1936) 230-263 · JFM 62.0136.03 · doi:10.2307/1968694
[41] [Si2] C.L. Siegel: On the theory of indefinite quadratic forms. Ann. of Math.45 (1944) 577-622 · Zbl 0063.07006 · doi:10.2307/1969191
[42] [Sp] N. Spaltenstein: On the number of rational points of homogeneous spaces over finite fields, preprint, May 1993
[43] [S] R. Steinberg: Endomorphisms of linear algebraic groups. Memoirs A.M.S.80 (1968) · Zbl 0164.02902
[44] [We1] A. Weil: Sur la théorie des formes quadratiques. Colloque sur la théorie des groupes algébriques, C.B.R.M., Bruxelles 1962, pp. 9-22
[45] [We2] A. Weil: Adeles and algebraic Groups. Boston: Birkhäuser 1982
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.