Prehomogeneous vector spaces and ergodic theory II

Witte, Dave; Yukie, Akihiko; Zierau, Roger

doi:10.1090/S0002-9947-99-02224-2

Prehomogeneous vector spaces and ergodic theory II
HTML articles powered by AMS MathViewer

by Dave Witte, Akihiko Yukie and Roger Zierau PDF

Trans. Amer. Math. Soc. 352 (2000), 1687-1708 Request permission

Abstract:

We apply M. Ratner’s theorem on closures of unipotent orbits to the study of three families of prehomogeneous vector spaces. As a result, we prove analogues of the Oppenheim Conjecture for simultaneous approximation by values of certain alternating bilinear forms in an even number of variables and certain alternating trilinear forms in six and seven variables.

References

Maple V. Waterloo Maple Inc., Waterloo, Ontario, 1994.
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
F. Reese Harvey, Spinors and calibrations, Perspectives in Mathematics, vol. 9, Academic Press, Inc., Boston, MA, 1990. MR 1045637
Jun-ichi Igusa, On a certain class of prehomogeneous vector spaces, J. Pure Appl. Algebra 47 (1987), no. 3, 265–282. MR 910424, DOI 10.1016/0022-4049(87)90051-X
Kable, A.C., and A. Yukie. Prehomogeneous vector spaces and field extensions II. Invent. Math. 130 (1997), 315–344.
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
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
David Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. With a section by G. M. Bergman. MR 0209285
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, 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 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
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
M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155. MR 430336
Mikio Sato and Takuro Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2) 100 (1974), 131–170. MR 344230, DOI 10.2307/1970844
Nimish A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann. 289 (1991), no. 2, 315–334. MR 1092178, DOI 10.1007/BF01446574
Takuro Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132–188. MR 289428, DOI 10.2969/jmsj/02410132
David J. Wright and Akihiko Yukie, Prehomogeneous vector spaces and field extensions, Invent. Math. 110 (1992), no. 2, 283–314. MR 1185585, DOI 10.1007/BF01231334
Klaus Langmann, Lösungsanzahl der homogenen Normformengleichung, Compositio Math. 94 (1994), no. 1, 29–49 (German). MR 1302310
Yukie, A. Prehomogeneous vector spaces and ergodic theory I. Duke Math. J. 90 (1997), 123–147.