Weakly commensurable arithmetic groups and isospectral locally symmetric spaces
Publications Mathématiques de l'IHÉS, Tome 109 (2009), pp. 113-184.
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     title = {Weakly commensurable arithmetic groups and isospectral locally symmetric spaces},
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     publisher = {Springer-Verlag},
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     url = {http://archive.numdam.org/articles/10.1007/s10240-009-0019-6/}
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Prasad, Gopal; Rapinchuk, Andreis. Weakly commensurable arithmetic groups and isospectral locally symmetric spaces. Publications Mathématiques de l'IHÉS, Tome 109 (2009), pp. 113-184. doi : 10.1007/s10240-009-0019-6. http://archive.numdam.org/articles/10.1007/s10240-009-0019-6/

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

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

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

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 | Zbl

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

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

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

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 | Zbl

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

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

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 | EuDML | MR | Zbl

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 | Zbl

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

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

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

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

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

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

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

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

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

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 | Zbl

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 | EuDML | MR | Zbl

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

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

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

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

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 | EuDML | Numdam | MR | Zbl

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

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

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

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 | Zbl

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

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 | Zbl

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 | Zbl

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

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

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

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

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

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

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

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

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 | Zbl

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

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

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

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

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