Weakly commensurable arithmetic groups and isospectral locally symmetric spaces
Prasad, Gopal ; Rapinchuk, Andreis.
Publications Mathématiques de l'IHÉS, Tome 110 (2009), p. 113-184 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{PMIHES_2009__109__113_0,
     author = {Prasad, Gopal and Rapinchuk, Andreis.},
     title = {Weakly commensurable arithmetic groups and isospectral locally symmetric spaces},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     volume = {110},
     year = {2009},
     pages = {113-184},
     doi = {10.1007/s10240-009-0019-6},
     mrnumber = {2511587},
     zbl = {1176.22011},
     language = {en},
     url = {http://dml.mathdoc.fr/item/PMIHES_2009__109__113_0}
}

Prasad, Gopal; Rapinchuk, Andreis. Weakly commensurable arithmetic groups and isospectral locally symmetric spaces. Publications Mathématiques de l'IHÉS, Tome 110 (2009) pp. 113-184. doi : 10.1007/s10240-009-0019-6. http://gdmltest.u-ga.fr/item/PMIHES_2009__109__113_0/

1. J. Ax, On Schanuel's conjecture, Ann. Math. (2) 93 (1971), p. 252-268 | MR 277482

2. A. Baker, Transcendental Number Theory, Cambridge Mathematical Library (1990), Cambridge Univ. Press, Cambridge | MR 1074572 | Zbl 0715.11032

3. A. Borel, Linear Algebraic Groups, GTM 126 (1991), Springer, Berlin | MR 1102012 | Zbl 0726.20030

4. A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. IHES 69 (1989), p. 119-171 | Numdam | MR 1019963 | Zbl 0707.11032

5. A. Borel, J. Tits, Groupes réductifs, Publ. Math. IHES 27 (1965), p. 55-151 | Numdam | MR 207712 | Zbl 0145.17402

6. M. V. Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math. J. 72 (1993), p. 217-239 | MR 1242885 | Zbl 0849.12011

7. N. Bourbaki, Groupes et algèbres de Lie, Hermann, Paris, Chap. III, 1972, Chap. IV-VI, 1968. | Zbl 0483.22001

8. T. Chinburg, E. Hamilton, D. D. Long, A. W. Reid, Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds, Duke Math. J. 145 (2008), p. 25-44 | MR 2451288 | Zbl 1169.53030

9. S. Debacker, Parametrizing conjugacy classes of maximal unramified tori via Bruhat-Tits theory, Michigan Math. J. 54 (2006), p. 157-178 | MR 2214792 | Zbl 1118.22005

10. J. J. Duistermaat, V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), p. 39-79 | MR 405514 | Zbl 0307.35071

11. J. J. Duistermaat, J. A. C. Kolk, V. S. Varadarajan, Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math. 52 (1979), p. 27-93 | MR 532745 | Zbl 0434.58019

12. E. V. Erovenko, A. S. Rapinchuk, Bounded generation of S-arithmetic subgroups of isotropic orthogonal groups over number fields, J. Number Theory 119 (2006), p. 28-48 | MR 2228948 | Zbl 1168.20020

13. R. Gangolli, The length spectra of some compact manifolds, J. Differ. Geom. 12 (1977), p. 403-424 | MR 650997 | Zbl 0365.53016

14. P. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, (1995), CRC Press, Boca Raton | MR 1396308 | Zbl 0856.58001

15. P. Gille, Type des tores maximaux des groupes semi-simples, J. Ramanujan Math. Soc. 19 (2004), p. 213-230 | MR 2139505 | Zbl 1193.20057

16. G. Harder, Bericht über neuere Resultate der Galoiskohomologie halbeinfacher Gruppen, Jber. Deutsch. Math.-Verein. 70 (1967), p. 182-216 | MR 242838 | Zbl 0194.05701

17. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, (2001), AMS, Providence | MR 1834454 | Zbl 0993.53002

18. K. Kariyama, On conjugacy classes of maximal tori in classical groups, J. Algebra 125 (1989), p. 133-149 | MR 1012667 | Zbl 0682.20031

19. C. J. Leininger, D. B. McReynolds, W. Neumann, and A. W. Reid, Length and eigenvalue equivalence, Int. Math. Res. Notices, 24 (2007). | MR 2377017 | Zbl 1158.53032

20. A. Lubotzky, B. Samuels, U. Vishne, Division algebras and noncommensurable isospectral manifolds, Duke Math. J. 135 (2006), p. 361-379 | MR 2267287 | Zbl 1123.58020

21. G. A. Margulis, Discrete Subgroups of Semi-Simple Lie Groups, (1991), Springer, Berlin | MR 1090825 | Zbl 0732.22008

22. H. P. Mckean, The Selberg trace formula as applied to a compact Riemann surface, Commun. Pure Appl. Math. 25 (1972), p. 225-246 | MR 473166 | Zbl 0317.30018

23. D. Morris, Bounded generation of SL(n,A) (after D. Carter, G. Keller, and E. Paige), N. Y. J. Math. 13 (2007), p. 383-421 | MR 2357719 | Zbl 1137.20045

24. A. Yu. Ol'Shanskii, Periodic quotient groups of hyperbolic groups, Math. USSR-Sb. 72 (1992), p. 519-541 | MR 1119008 | Zbl 0820.20044

25. R. S. Pierce, Associative Algebras, GTM 88 (1982), Springer, Berlin | MR 674652 | Zbl 0497.16001

26. V. P. Platonov, A. S. Rapinchuk, Algebraic Groups and Number Theory, (1994), Academic Press, New York | MR 1278263 | Zbl 0841.20046

27. G. Prasad, Lattices in semi-simple groups over local fields, in: Adv. in Math. Studies in Algebra and Number Theory, (1979), Academic Press, New York | MR 535769

28. G. Prasad, Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bull. Soc. Math. France 110 (1982), p. 197-202 | Numdam | MR 667750 | Zbl 0492.20029

29. G. Prasad, M. S. Raghunathan, Cartan subgroups and lattices in semi-simple groups, Ann. Math. (2) 96 (1972), p. 296-317 | MR 302822 | Zbl 0245.22013

30. G. Prasad, A. S. Rapinchuk, Computation of the metaplectic kernel, Publ. Math. IHES 84 (1996), p. 91-187 | Numdam | MR 1441007 | Zbl 0941.22019

31. G. Prasad, A. S. Rapinchuk, Irreducible tori in semisimple groups, Int. Math. Res. Notices 23 (2001), p. 1229-1242 | MR 1866442 | Zbl 1057.22025

32. G. Prasad, A. S. Rapinchuk, Existence of irreducible ℝ-regular elements in Zariski-dense subgroups, Math. Res. Lett. 10 (2003), p. 21-32 | MR 1960120 | Zbl 1029.22020

33. G. Prasad, A. S. Rapinchuk, Zariski-dense subgroups and transcendental number theory, Math. Res. Lett. 12 (2005), p. 239-249 | MR 2150880 | Zbl 1072.22009

34. G. Prasad, A. S. Rapinchuk, On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior, Adv. Math. 207 (2006), p. 646-660 | MR 2271021 | Zbl 1132.20024

35. G. Prasad and A. S. Rapinchuk, Local-global principles for embedding of fields with involution into simple algebras with involution, available at: arXiv:0806.0596 (to appear in Comment. Math. Helv.). | MR 2653693 | Zbl 1223.11047

36. M. S. Raghunathan, Discrete Subgroups of Lie Groups, (1972), Springer, Berlin | MR 507234 | Zbl 0254.22005

37. M. S. Raghunathan, Tori in quasi-split groups, J. Ramanujan Math. Soc. 19 (2004), p. 281-287 | MR 2125504 | Zbl 1080.20042

38. A. W. Reid, Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds, Duke Math. J. 65 (1992), p. 215-228 | MR 1150584 | Zbl 0776.58040

39. J.-P. Serre, Galois Cohomology, (1997), Springer, Berlin | MR 1466966 | Zbl 0902.12004

40. D. J. Saltman, A note on quaternion algebras (preprint).

41. T. A. Springer, Linear Algebraic Groups, (1998), Birkhäuser, Basel | MR 1642713 | Zbl 1202.20048

42. T. Sunada, Riemann coverings and isospectral manifolds, Ann. Math. (2) 121 (1985), p. 169-186 | MR 782558 | Zbl 0585.58047

43. J. Tits, Algebraic and abstract simple groups, Ann. Math. (2) 80 (1964), p. 313-329 | MR 164968 | Zbl 0131.26501

44. J. Tits, Classification of algebraic semisimple groups, in: Algebraic Groups and Discontinuous Groups, Proc. Sympos. Pure Math. 9 (1966), Amer. Math. Soc., Providence | MR 224710 | Zbl 0238.20052

45. M.-F. Vignéras, Variétés Riemanniennes isospectrales et non isométriques, Ann. Math. (2) 112 (1980), p. 21-32 | Zbl 0445.53026

46. E. B. Vinberg, Rings of definition of dense subgroups of semisimple linear groups, Math. USSR Izv. 5 (1971), p. 45-55 | MR 279206 | Zbl 0252.20043

47. V. E. Voskresenskii, Algebraic Groups and Their Birational Invariants, (1998), AMS, Providence | MR 1634406 | Zbl 0974.14034

48. B. Weisfeiler, Strong approximation for Zariski-dense subgroups of semi-simple algebraic groups, Ann. Math. (2) 120 (1984), p. 271-315 | MR 763908 | Zbl 0568.14025