|
Values of Indefinite Quadratic Forms at Integral Points and Flows on Spaces of Lattices
Author(s):
Armand
Borel
Journal:
Bull. Amer. Math. Soc.
32
(1995),
184-204.
MSC (1991):
Primary 22-02, 73K12
MathSciNet review:
1302785
Retrieve article in:
PDF
References |
Similar articles |
Additional information
References:
- [BB]
- F.~Bien and A.~Borel, Sous-groupes \'epimorphiques de groupes alg\'ebriques lin\'eaires. \RM I, C.~R. ~Acad.~Sci. Paris S\'er. I Math. \textbf{315} (1992), 649--653. MR 1183796
- [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. \textbf{47} (1992), 118--120.
- [BP1]
- A. Borel and G. Prasad, Valeurs de formes quadratiques aux points entiers, C.R. Acad. Sci. Paris S\'er. I Math. \textbf{307} (1988), 217--220. MR 956809
- [BP2]
- A. Borel and G. Prasad, Values of isotropic quadratic forms at $S$-integral points, Compositio Math. \textbf{83} (1992), 347--372. MR 1175945
- [BT]
- A. Borel and G. Prasad, Homomorphismes {\rm ``}abstraits{\rm ''} de groupes alg\`ebriques simples, Ann. of Math. (2) \textbf{97} (1973), 499--571. MR 316587
- [Bu]
- N.~Bourbaki, Int\'egration, Chap. 7, 8, Hermann, Paris, 1963. MR 179291
- [Bu1]
- N.~Bourbaki, Groupes et alg\`ebres de Lie, Chap. 2, 3, Hermann, Paris, 1972. MR 573068
- [C]
- S.~Chowla, A theorem on irrational indefinite quadratic forms, J. London Math. Soc. \textbf{9} (1934), 162--163.
- [D1]
- S. G. Dani, Invariant measures of horospherical flows on non-compact homogeneous spaces, Invent. Math. \textbf{47} (1978), 101--138. MR 578655
- [D2]
- S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math. \textbf{64} (1981), 357--385. MR 629475
- [D3]
- S. G. Dani, Orbits of horospherical flows, Duke Math. J. \textbf{53} (1986), 177--188. MR 835804
- [DM1]
- S. G. Dani and G. A. Margulis, Values of quadratic forms at primitive integral points, Invent. Math. \textbf{98} (1989), 405--424. MR 1016271
- [DM2]
- S. G. Dani and G. A. Margulis, Orbit closures of generic unipotent flows on homogeneous spaces of $SL(3, {\R })$, Math. Ann. \textbf{286} (1990), 101--128. MR 1032925
- [DM3]
- S. G. Dani and G. A. Margulis, Values of quadratic forms at integral points\,\RM : An elementary approach, Enseign. Math. (2) \textbf{36} (1990), 143--174. MR 1071418
- [DM4]
- S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, Adv. Soviet Math. \textbf{6} (1993), 91--137. MR 1237827
- [DS]
- S. G. Dani and J. Smillie, Uniform distribution of horocyclic orbits for Fuchsian groups, Duke Math. J. \textbf{51} (1984), 185--194. MR 744294
- [DH]
- H. Davenport and H. Heilbronn, On indefinite quadratic in five variables, J. London Math. Soc. \textbf{21} (1946), 185--193. MR 20578
- [DR]
- H. Davenport and H. Ridout, Indefinite quadratic forms, Proc. London Math. Soc. \textbf{9} (1959), 544--555. MR 109140
- [F]
- H. Furstenberg, \emph{The unique ergodicity of the horocycle flow}.
- [G]
- E.~Ghys, Dynamique des flots unipotents sur les espaces homog\`enes, Ast\'erisque {\bf 206} (S\'em. Bourbaki 1992--93, no. 747) (1992), 93--136. MR 1206065
- [H]
- G. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J. \textbf{2} (1936), 530--542. MR 1545946
- [L]
- D.~J.~Lewis, \emph{The distribution of values of real quadratic forms at integer points}, Amer. Math. Soc., Providence, RI, 1973 pp.~159--174. MR 337764
- [M1]
- G.A. Margulis, \emph{Lie groups and ergodic theory}.
- [M2]
- G.A. Margulis, \emph{Indefinite quadratic forms and unipotent flows on homogeneous spaces}, Polish Scientific Publishers, Warsaw, 1989. MR 1102736
- [M3]
- G.A. Margulis, \emph{Discrete subgroups and ergodic theory}.
- [MT1]
- G. A. Margulis and G. Tomanov, Measure rigidity for algebraic groups over local fields, C. R. Acad. Sci. Paris S\'er. I. Math. \textbf{315} (1992), 1221--1226. MR 1194522
- [MT2]
- G. A. Margulis and G. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. \textbf{116} (1994), 347--392. MR 1253197
- [M]
- G.~D.~Mostow, Homogeneous spaces with finite invariant measure, Ann. of Math. \textbf{75} (1992), 17--37. MR 145007
- [Mo]
- S.~Mozes, Epimorphic subgroups and invariant measures, preprint. MR 1366316
- [MS]
- S.~Mozes and N.~Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory and Dynamical Systems. MR 1314973
- [O1]
- A. Oppenheim, The minima of indefinite quaternary quadratic forms of signature $0$, Proc. Nat. Acad. Sci. U.S.A \textbf{15} (1929), 724--727.
- [O2]
- A. Oppenheim, The minima of indefinite quaternary quadratic forms, Ann. of Math. \textbf{32} (1931), 271--298. MR 1502997
- [O3]
- A. Oppenheim, Values of quadratic forms. \RM I, Quart. J. Math. Oxford Ser. (2) \textbf{4} (1953), 54--59. MR 54650
- [O4]
- A. Oppenheim, Values of quadratic forms. {\rm II}, Quart. J. Math. Oxford Ser. (2) \textbf{4} (1953), 60--66. MR 54651
- [O5]
- A. Oppenheim, Values of quadratic forms. {\rm III}, Monatsh. Math. \textbf{57} (1953), 97--101. MR 56648
- [P]
- W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. \textbf{91} (1969), 757--771. MR 260975
- [Pr]
- G.~Prasad, Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bull. Soc. Math. France \textbf{110} (1982), 197--202. MR 667750
- [RR]
- S. Raghavan and K.G. Ramanathan, On a diophantine inequality concerning quadratic forms, Nachr. Akad. Wiss. G\"ottingen Math.-Phys. Kl. II (1968), 251--262. MR 263743
- [R]
- M. S.~Raghunathan, \emph{Discrete subgroups of Lie groups}, Springer-Verlag, Berlin, 1992. MR 507234
- [R1]
- M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math. \textbf{101} (1990), 449--482. MR 1062971
- [R2]
- M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math. \textbf{165} (1990), 229--309. MR 1075042
- [R3]
- M. Ratner, On Raghunathan{\rm '}s measure conjecture, Ann. of Math. \textbf{134} (1991), 545--607. MR 1135878
- [R4]
- M. Ratner, Raghunathan{\rm '}s topological conjecture and distributions of unipotent flows, Duke Math. J. \textbf{63} (1991), 235--280. MR 1106945
- [R5]
- M. Ratner, Raghunathan{\rm '}s conjectures for $\BSL _2(\R )$, Israel J. Math. \textbf{80} (1992), 1--31. MR 1248925
- [R6]
- M. Ratner, Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal. \textbf{4} (1994), 236--257. MR 1262705
- [R7]
- M. Ratner, Raghunathan{\rm '}s conjectures for $p$-adic Lie groups, Internat. Math. Res. Notices \textbf{5} (1993), 141--146. MR 1219864
- [R8]
- M. Ratner, Raghunathan{\rm '}s conjectures for cartesian products of real and $p$-adic groups, Duke Math. J.
- [R9]
- M. Ratner, Interactions between ergodic theory, Lie groups and number theory, Proc. ICM 94.
- [S]
- J-P.~Serre, \emph{A course in arithmetic}, Springer, New York, 1973. MR 344216
- [S1]
- J-P.~Serre, \emph{Lie algebras and Lie groups \rm (second ed.)}, Springer, New York, 1992. MR 1176100
- [Sh]
- N.A.~Shah, Limit distributions of polynomial trajectories on homogeneous spaces, preprint. MR 1291701
Similar Articles:
Retrieve articles in Bulletin of the American Mathematical Society
with MSC
(1991):
22-02, 73K12
Retrieve articles in all Journals with MSC
(1991):
22-02, 73K12
Additional Information:
DOI:
10.1090/S0273-0979-1995-00587-2
PII:
S 0273-0979(1995)00587-2
|
