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Values of indefinite quadratic forms at integral points and flows on spaces of lattices


Author: Armand Borel
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
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  • [BB] F. Bien and A. Borel, Sous-groupes épimorphiques de groupes algébriques linéaires. I, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 649-653. MR 1183796 (93i:20048)
  • [B] 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.
  • [BP1] A. Borel and G. Prasad, Valeurs de formes quadratiques aux points entiers, C.R. Acad. Sci. Paris Sér. I Math. 307 (1988), 217-220. MR 956809 (90j:11031)
  • [BP2] -, Values of isotropic quadratic forms at S-integral points, Compositio Math. 83 (1992), 347-372. MR 1175945 (93j:11022)
  • [BT] -, Homomorphismes "abstraits" de groupes algèbriques simples, Ann. of Math. (2) 97 (1973), 499-571. MR 0316587 (47:5134)
  • [Bu] N. Bourbaki, Intégration, Chap. 7, 8, Hermann, Paris, 1963.
  • [Bu1] -, Groupes et algèbres de Lie, Chap. 2, 3, Hermann, Paris, 1972.
  • [C] S. Chowla, A theorem on irrational indefinite quadratic forms, J. London Math. Soc. 9 (1934), 162-163.
  • [D1] S. G. Dani, Invariant measures of horospherical flows on non-compact homogeneous spaces, Invent. Math. 47 (1978), 101-138. MR 0578655 (58:28260)
  • [D2] -, Invariant measures and minimal sets of horospherical flows, Invent. Math. 64 (1981), 357-385. MR 629475 (83c:22009)
  • [D3] -, Orbits of horospherical flows, Duke Math. J. 53 (1986), 177-188. MR 835804 (87i:22026)
  • [DM1] S. G. Dani and G. A. Margulis, Values of quadratic forms at primitive integral points, Invent. Math. 98 (1989), 405-424. MR 1016271 (90k:22013b)
  • [DM2] -, Orbit closures of generic unipotent flows on homogeneous spaces of $ {SL(3,{\textbf{R}})}$, Math. Ann. 286 (1990), 101-128. MR 1032925 (91k:22026)
  • [DM3] -, Values of quadratic forms at integral points: An elementary approach, Enseign. Math. (2) 36 (1990), 143-174. MR 1071418 (91k:11053)
  • [DM4] -, Limit distributions of orbits of unipotent flows and values of quadratic forms, Adv. Soviet Math. 6 (1993), 91-137. MR 1237827 (95b:22024)
  • [DS] S. G. Dani and J. Smillie, Uniform distribution of horocyclic orbits for Fuchsian groups, Duke Math. J. 51 (1984), 185-194. MR 744294 (85f:58093)
  • [DH] H. Davenport and H. Heilbronn, On indefinite quadratic in five variables, J. London Math. Soc. 21 (1946), 185-193. MR 0020578 (8:565e)
  • [DR] H. Davenport and H. Ridout, Indefinite quadratic forms, Proc. London Math. Soc. 9 (1959), 544-555. MR 0109140 (22:28)
  • [F] H. Furstenberg, The unique ergodicity of the horocycle flow, Recent Advances in Topological Dynamics, Lecture Notes in Math., vol. 318, Springer, New York, 1972, 95-115. MR 0393339 (52:14149)
  • [G] E. Ghys, Dynamique des flots unipotents sur les espaces homogènes, Astérisque 206 (Sém. Bourbaki 1992-93, no. 747) (1992), 93-136. MR 1206065 (94e:58101)
  • [H] G. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J. 2 (1936), 530-542. MR 1545946
  • [L] D.J. Lewis, The distribution of values of real quadratic forms at integer points, Proc. Sympos. Pure Math., vol. XXIV, Amer. Math. Soc., Providence, RI, 1973, pp. 159-174. MR 0337764 (49:2533)
  • [M1] G.A. Margulis, Lie groups and ergodic theory, Algebra Some Current Trends (L. L. Avramov, ed.) Proc. Varna 1986, Lecture Notes in Math., vol. 1352, Springer, New York, 130-146. MR 981823 (91a:22009)
  • [M2] -, Indefinite quadratic forms and unipotent flows on homogeneous spaces, Banach Center Publ., vol. 23, Polish Scientific Publishers, Warsaw, 1989.
  • [M3] -, Discrete subgroups and ergodic theory, Number Theory, Trace Formulas and Discrete Groups (symposium in honour of A. Selberg), Academic Press, San Diego, CA, 1989, pp. 377-398. MR 993307 (89m:11003)
  • [MT1] G. A. Margulis and G. Tomanov, Measure rigidity for algebraic groups over local fields, C. R. Acad. Sci. Paris Sér. I. Math. 315 (1992), 1221-1226. MR 1194522 (94f:22016)
  • [MT2] -, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. 116 (1994), 347-392. MR 1253197 (95k:22013)
  • [M] G. D. Mostow, Homogeneous spaces with finite invariant measure, Ann. of Math. 75 (1992), 17-37. MR 0145007 (26:2546)
  • [Mo] S. Mozes, Epimorphic subgroups and invariant measures, preprint. MR 1366316 (96m:58143)
  • [MS] S. Mozes and N. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory and Dynamical Systems (to appear). MR 1314973 (95k:58096)
  • [O1] A. Oppenheim, The minima of indefinite quaternary quadratic forms of signature 0, Proc. Nat. Acad. Sci. U.S.A 15 (1929), 724-727.
  • [O2] -, The minima of indefinite quaternary quadratic forms, Ann. of Math. 32 (1931), 271-298. MR 1502997
  • [O3] -, Values of quadratic forms. I, Quart. J. Math. Oxford Ser. (2) 4 (1953), 54-59. MR 0054650 (14:955a)
  • [O4] -, Values of quadratic forms. II, Quart. J. Math. Oxford Ser. (2) 4 (1953), 60-66. MR 0054651 (14:955b)
  • [O5] -, Values of quadratic forms. III, Monatsh. Math. 57 (1953), 97-101. MR 0056648 (15:106e)
  • [P] W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757-771. MR 0260975 (41:5595)
  • [Pr] G. Prasad, Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bull. Soc. Math. France 110 (1982), 197-202. MR 667750 (83m:20064)
  • [RR] S. Raghavan and K.G. Ramanathan, On a diophantine inequality concerning quadratic forms, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1968), 251-262. MR 0263743 (41:8343)
  • [R] M. S. Raghunathan, Discrete subgroups of Lie groups, Ergeb. Math. Grenzgeb. (3), vol. 68, Springer-Verlag, Berlin, 1992. MR 0507234 (58:22394a)
  • [R1] M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math. 101 (1990), 449-482. MR 1062971 (92h:22015)
  • [R2] -, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math. 165 (1990), 229-309. MR 1075042 (91m:57031)
  • [R3] -, On Raghunathan's measure conjecture, Ann. of Math. 134 (1991), 545-607. MR 1135878 (93a:22009)
  • [R4] -, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), 235-280. MR 1106945 (93f:22012)
  • [R5] -, Raghunathan's conjectures for $ {{\textbf{S}}{{\textbf{L}}_2}({\textbf{R}})}$, Israel J. Math. 80 (1992), 1-31. MR 1248925 (94k:22024)
  • [R6] -, Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal. 4 (1994), 236-257. MR 1262705 (95c:22018)
  • [R7] -, Raghunathan's conjectures for p-adic Lie groups, Internat. Math. Res. Notices 5 (1993), 141-146.
  • [R8] -, Raghunathan's conjectures for cartesian products of real and p-adic groups, Duke Math. J. (to appear).
  • [R9] -, Interactions between ergodic theory, Lie groups and number theory, Proc. ICM 94 (to appear).
  • [S] J-P. Serre, A course in arithmetic, Graduate Texts in Math., vol. 7, Springer, New York, 1973. MR 0344216 (49:8956)
  • [S1] -, Lie algebras and Lie groups (second ed.), Lecture Notes in Math., vol. 1500, Springer, New York, 1992. MR 1176100 (93h:17001)
  • [Sh] N.A. Shah, Limit distributions of polynomial trajectories on homogeneous spaces, preprint. MR 1291701 (95j:22022)

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DOI: https://doi.org/10.1090/S0273-0979-1995-00587-2
Article copyright: © Copyright 1995 American Mathematical Society

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