Summary
We are interested in counting integer and rational points in affine algebraic varieties, also under congruence conditions. We introduce the notions of a strongly Hardy-Littlewood variety and a relatively Hardy-Littlewood variety, in terms of counting rational points satisfying congruence conditions. The definition of a strongly Hardy-Littlewood variety is given in such a way that varieties for which the Hardy-Littlewood circle method is applicable are strongly Hardy-Littlewood.
We prove that certain affine homogeneous spaces of semisimple groups are strongly Hardy-Littlewood varieties. Moreover, we prove that many homogeneous spaces are relatively Hardy-Littlewood, but not strongly Hardy-Littlewood. This yields a new class of varieties for with the asymptotic density of integer points can be computed in terms of a product of local densities.
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References
[Bi] B.J. Birch: Forms in many variables. Proc. Roy. Soc. Ser. A265 (1962) 245–263
[B-HC] A. Borel, Harish-Chandra: Arithmetic subgroups of algebraic groups. Ann. Math.75 (1962) 485–535
[Brl] A. Borel: Some finiteness properties of adele groups over number fields. Publ. Math.16 (1963) 101–126
[Bo1] M.V. Borovoi: On strong approximation for homogeneous spaces. Doklady Akad. Nauk BSSR33 (1989) 293–296 (Russian)
[Bo2] M.V. Borovoi: The algebraic fundamental group and abelian Galois cohomology of reductive algebraic groups, Preprint, MPI/89-90, Bonn
[Bo3] M.V. Borovoi: On weak approximation in homogeneous spaces of algebraic groups. Soviet Math. Doklady42 (1991) 247–251
[Bo4] M.V. Borovoi: The Hasse principle for homogeneous spaces. J. Reine Angew. Math.426 (1992) 179–192
[Bo5] M.V. Borovoi: Abelianization of the second nonabelian Galois cohomology. Duke Math. J.72 (1993) 217–239
[Ca] J.W.S. Cassels: Rational Quadratic Forms. London: Academic Press, 1978
[CS] J.H. Conway, N.J.A. Sloane: Sphere Packings, Lattices and Groups, 2nd edn. New York: Springer 1993
[Da] H. Davenport: Analytic Methods for Diophantine Equations and Diophantine Inequalities. Ann Arbor, Michigan: Ann Arbor Publishers, 1962
[Di] J. Dixmier: Quelques aspects de la théorie des invariants. Gaz. Mathematiciens43 (1990) 39–64
[DRS] W. Duke, Z. Rudnick, P. Sarnak: Density of integer points on affine homogeneous varieties. Duke Math. J.71 (1993) 143–179
[EM] A. Eskin, C. McMullen: Mixing, counting, and equidistribution in Lie groups. Duke Math. J.71 (1993) 181–209
[EMS] A. Eskin, S. Mozes, N. Shah: Unipotent flows and counting lattice points on homogeneous spaces, Preprint
[ERS] A. Eskin, Z. Rudnick, P. Sarnak: A proof of Siegel's weight formula. Duke Math. J., Int. Math. Res. Notices5 (1991) 65–69
[Esk] A. Eskin: Ph. D. Thesis, Princeton University 1993
[Es] T. Estermann: A new application of the Hardy-Littlewood-Kloosterman method. Proc. London Math. Soc.12 (1962) 425–444
[FMT] J. Franke, Yu. I. Manin, Yu. Tschinkel: Rational points of bounded height on Fano varieties. Invent. Math.95 (1989) 421–435
[Ha] G. Harder: Über die Galoiskohomologie halbeinfacher Matrizengruppen, I, Math. Z. 90 (1965) 404–428; II, Math. Z.92 (1966) 396–415
[HB] D.R. Heath-Brown: The density of zeros of forms for which weak approximation fails. Math. Comp.59 (1992) 613–623
[Ig] J.-I. Igusa: Lectures on Forms of Higher Degree. Bombay: Tata Institute of Fundamental Research 1978
[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
[Kn2] M. Kneser: Starke Approximation in algebraischen Gruppen, I. J. Reine Angew. Math.218 (1965) 190–203
[Ko1] R.E. Kottwitz: Stable trace formula: cuspidal tempered terms. Duke Math. J.51 (1984) 611–650
[Ko2] R.E. Kottwitz: Stable trace formula: elliptic singular terms. Math. Ann.275 (1986) 365–399
[Ko3] R.E. Kottwitz: Tamagawa numbers. Ann. Math.127 (1988) 629–646
[La] S. Lang: Algebraic groups over finite fields. Am. J. Math.78 (1956) 555–563
[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
[Min] Kh.P. Minchev: Strong approximation for varieties over an algebraic number field. Doklady Akad. Nauk BSSR33 (1989) 5–8 (Russian)
[O] T. Ono: On the relative theory of Tamagawa numbers. Ann. Math.82 (1965) 88–111
[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
[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
[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)
[Ro] M. Rosenlicht: Toroidal algebraic groups. Proc. A.M.S.12 (1961) 984–988
[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
[Sch] W.M. Schmidt: The density of integer points on homogeneous varieties. Acta. Math.154 (1985) 243–296
[Se1] J.-P. Serre: Cohomologie galoisienne. (Lect. Notes Math. vol. 5) Berlin Heidelberg New York: Springer 1965
[Se2] J.-P. Serre: Resumés des cours de 1981–1982, 1982–1983 (Euvres, pp. 649–657, 669–674) Berlin Heidelberg New York: Springer 1986
[Sie1] C.L. Siegel: Über die analytische Theorie der quadratischen Formen II. Ann. Math.37 (1936) 230–263
[Si2] C.L. Siegel: On the theory of indefinite quadratic forms. Ann. of Math.45 (1944) 577–622
[Sp] N. Spaltenstein: On the number of rational points of homogeneous spaces over finite fields, preprint, May 1993
[S] R. Steinberg: Endomorphisms of linear algebraic groups. Memoirs A.M.S.80 (1968)
[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
[We2] A. Weil: Adeles and algebraic Groups. Boston: Birkhäuser 1982
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Oblatum 15-IX-1993 & 31-I-1994
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Borovoi, M., Rudnick, Z. Hardy-Littlewood varieties and semisimple groups. Invent Math 119, 37–66 (1995). https://doi.org/10.1007/BF01245174
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DOI: https://doi.org/10.1007/BF01245174