Values of indefinite quadratic forms at integral points and flows on spaces of lattices
HTML articles powered by AMS MathViewer
- by Armand Borel PDF
- Bull. Amer. Math. Soc. 32 (1995), 184-204 Request permission
References
- Frédéric Bien and Armand Borel, Sous-groupes épimorphiques de groupes linéaires algébriques. I, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 6, 649–653 (French, with English and French summaries). MR 1183796 A. Borel, Values of quadratic forms at S-integral points, Algebraic Groups and Number Theory (V. Platonov and A. S. Rapinchuk, eds.), Uspekhi Mat. Nauk. 47 (1992), 118-120; Russian Math. Surveys 47 (1992), 134-136.
- Armand Borel and Gopal Prasad, Valeurs de formes quadratiques aux points entiers, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 6, 217–220 (French, with English summary). MR 956809
- Armand Borel and Gopal Prasad, Values of isotropic quadratic forms at $S$-integral points, Compositio Math. 83 (1992), no. 3, 347–372. MR 1175945
- Armand Borel and Jacques Tits, Homomorphismes “abstraits” de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499–571 (French). MR 316587, DOI 10.2307/1970833 N. Bourbaki, Intégration, Chap. 7, 8, Hermann, Paris, 1963. —, Groupes et algèbres de Lie, Chap. 2, 3, Hermann, Paris, 1972. S. Chowla, A theorem on irrational indefinite quadratic forms, J. London Math. Soc. 9 (1934), 162-163.
- S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math. 47 (1978), no. 2, 101–138. MR 578655, DOI 10.1007/BF01578067
- S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math. 64 (1981), no. 2, 357–385. MR 629475, DOI 10.1007/BF01389173
- S. G. Dani, Orbits of horospherical flows, Duke Math. J. 53 (1986), no. 1, 177–188. MR 835804, DOI 10.1215/S0012-7094-86-05312-3
- G. A. Margulis, Discrete subgroups and ergodic theory, Number theory, trace formulas and discrete groups (Oslo, 1987) Academic Press, Boston, MA, 1989, pp. 377–398. MR 993328
- S. G. Dani and G. A. Margulis, Orbit closures of generic unipotent flows on homogeneous spaces of $\textrm {SL}(3,\textbf {R})$, Math. Ann. 286 (1990), no. 1-3, 101–128. MR 1032925, DOI 10.1007/BF01453567
- S. G. Dani and G. A. Margulis, Values of quadratic forms at integral points: an elementary approach, Enseign. Math. (2) 36 (1990), no. 1-2, 143–174. MR 1071418
- S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 91–137. MR 1237827
- S. G. Dani and John Smillie, Uniform distribution of horocycle orbits for Fuchsian groups, Duke Math. J. 51 (1984), no. 1, 185–194. MR 744294, DOI 10.1215/S0012-7094-84-05110-X
- H. Davenport and H. Heilbronn, On indefinite quadratic forms in five variables, J. London Math. Soc. 21 (1946), 185–193. MR 20578, DOI 10.1112/jlms/s1-21.3.185
- H. Davenport and D. Ridout, Indefinite quadratic forms, Proc. London Math. Soc. (3) 9 (1959), 544–555. MR 109140, DOI 10.1112/plms/s3-9.4.544
- Harry Furstenberg, The unique ergodicity of the horocycle flow, Recent advances in topological dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., Vol. 318, Springer, Berlin, 1973, pp. 95–115. MR 0393339
- Étienne Ghys, Dynamique des flots unipotents sur les espaces homogènes, Astérisque 206 (1992), Exp. No. 747, 3, 93–136 (French, with French summary). Séminaire Bourbaki, Vol. 1991/92. MR 1206065
- Gustav A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J. 2 (1936), no. 3, 530–542. MR 1545946, DOI 10.1215/S0012-7094-36-00246-6
- D. J. Lewis, The distribution of the values of real quadratic forms at integer points, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 159–174. MR 0337764
- G. A. Margulis, Lie groups and ergodic theory, Algebra—some current trends (Varna, 1986) Lecture Notes in Math., vol. 1352, Springer, Berlin, 1988, pp. 130–146. MR 981823, DOI 10.1007/BFb0082022 —, Indefinite quadratic forms and unipotent flows on homogeneous spaces, Banach Center Publ., vol. 23, Polish Scientific Publishers, Warsaw, 1989.
- Karl Egil Aubert, Enrico Bombieri, and Dorian Goldfeld (eds.), Number theory, trace formulas and discrete groups, Academic Press, Inc., Boston, MA, 1989. Symposium in honor of Atle Selberg held at the University of Oslo, Oslo, June 14–20, 1987. MR 993307
- Gregori Aleksandrovitch Margulis and Georges Metodiev Tomanov, Measure rigidity for algebraic groups over local fields, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 12, 1221–1226 (English, with English and French summaries). MR 1194522
- G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. 116 (1994), no. 1-3, 347–392. MR 1253197, DOI 10.1007/BF01231565
- G. D. Mostow, Homogeneous spaces with finite invariant measure, Ann. of Math. (2) 75 (1962), 17–37. MR 145007, DOI 10.2307/1970416
- Shahar Mozes, Epimorphic subgroups and invariant measures, Ergodic Theory Dynam. Systems 15 (1995), no. 6, 1207–1210. MR 1366316, DOI 10.1017/S0143385700009871
- Shahar Mozes and Nimish Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 149–159. MR 1314973, DOI 10.1017/S0143385700008282 A. Oppenheim, The minima of indefinite quaternary quadratic forms of signature 0, Proc. Nat. Acad. Sci. U.S.A 15 (1929), 724-727.
- Alexander Oppenheim, The minima of indefinite quaternary quadratic forms, Ann. of Math. (2) 32 (1931), no. 2, 271–298. MR 1502997, DOI 10.2307/1968191
- A. Oppenheim, Values of quadratic forms. I, Quart. J. Math. Oxford Ser. (2) 4 (1953), 54–59. MR 54650, DOI 10.1093/qmath/4.1.54
- A. Oppenheim, Values of quadratic forms. II, Quart. J. Math. Oxford Ser. (2) 4 (1953), 60–66. MR 54651, DOI 10.1093/qmath/4.1.60
- A. Oppenheim, Value of quadratic forms. III, Monatsh. Math. 57 (1953), 97–101. MR 56648, DOI 10.1007/BF01299625
- William Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771. MR 260975, DOI 10.2307/2373350
- Gopal Prasad, Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bull. Soc. Math. France 110 (1982), no. 2, 197–202 (English, with French summary). MR 667750, DOI 10.24033/bsmf.1959
- S. Raghavan and K. G. Ramanathan, On a Diophantine inequality concerning quadratic forms, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1968 (1968), 251–262. MR 263743
- M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
- Marina Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math. 101 (1990), no. 2, 449–482. MR 1062971, DOI 10.1007/BF01231511
- Marina Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math. 165 (1990), no. 3-4, 229–309. MR 1075042, DOI 10.1007/BF02391906
- Marina Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), no. 3, 545–607. MR 1135878, DOI 10.2307/2944357
- Marina Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235–280. MR 1106945, DOI 10.1215/S0012-7094-91-06311-8
- Marina Ratner, Raghunathan’s conjectures for $\textrm {SL}(2,\mathbf R)$, Israel J. Math. 80 (1992), no. 1-2, 1–31. MR 1248925, DOI 10.1007/BF02808152
- M. Ratner, Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal. 4 (1994), no. 2, 236–257. MR 1262705, DOI 10.1007/BF01895839 —, Raghunathan’s conjectures for p-adic Lie groups, Internat. Math. Res. Notices 5 (1993), 141-146. —, Raghunathan’s conjectures for cartesian products of real and p-adic groups, Duke Math. J. (to appear). —, Interactions between ergodic theory, Lie groups and number theory, Proc. ICM 94 (to appear).
- J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216, DOI 10.1007/978-1-4684-9884-4
- Jean-Pierre Serre, Lie algebras and Lie groups, 2nd ed., Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 1992. 1964 lectures given at Harvard University. MR 1176100, DOI 10.1007/978-3-540-70634-2
- Nimish A. Shah, Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J. 75 (1994), no. 3, 711–732. MR 1291701, DOI 10.1215/S0012-7094-94-07521-2
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 32 (1995), 184-204
- MSC: Primary 22E40; Secondary 11H50, 11H55, 22-02
- DOI: https://doi.org/10.1090/S0273-0979-1995-00587-2
- MathSciNet review: 1302785