Geodesic
Geometric Spectra and Commensurability
Canadian journal of mathematics, Tome 67 (2015) no. 1, pp. 184-197

Voir la notice de l'article provenant de la source Cambridge University Press

The work of Reid, Chinburg–Hamilton–Long–Reid, Prasad–Rapinchuk, and the authorwith Reid have demonstrated that geodesics or totally geodesic submanifolds can sometimes be used to determine the commensurability class of an arithmetic manifold. The main results of this article show that generalizations of these results to other arithmetic manifolds will require a wide range of data. Specifically, we prove that certain incommensurable arithmetic manifolds arising from the semisimple Lie groups of the form ${{\left( \text{SL(d,}\,\mathbf{R}) \right)}^{r}}\,\times \,{{\left( \text{SL(d,}\,\mathbf{C}) \right)}^{s}}$ have the same commensurability classes of totally geodesic submanifolds coming from a fixed field. This construction is algebraic and shows the failure of determining, in general, a central simple algebra from subalgebras over a fixed field. This, in turn, can be viewed in terms of forms of $\text{S}{{\text{L}}_{d}}$ and the failure of determining the form via certain classes of algebraic subgroups.
DOI : 10.4153/CJM-2014-003-9
Mots-clés : 20G25, arithmetic groups, Brauer groups, arithmetic equivalence, locally symmetric manifolds 
McReynolds, D. B. Geometric Spectra and Commensurability. Canadian journal of mathematics, Tome 67 (2015) no. 1, pp. 184-197. doi: 10.4153/CJM-2014-003-9
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[1] [1] Aka, M., Arithmetic groups with isomorphic finite quotients. J. Algebra 352(2012), 322–340. Google Scholar | DOI

[2] [2] Borel, A. and Harish-Chandra, , Arithmetic subgroups of algebraic groups. Ann. of Math. (2)75(1962), 485–535. Google Scholar | DOI

[3] [3] Cassels, J.W. S. and Frölich, A., Algebraic number theory. London Mathematical Society, 1967. Google Scholar

[4] [4] Chernousov, V. I., Rapinchuk, A. S., and Rapinchuk, I. A., On the genus of a division algebra. C. R. Math. Acad. Sci. Paris 350(2012), 17–18. Google Scholar | DOI

[5] [5] Chinburg, T., Hamilton, E., Long, D. D., and Reid, A.W., Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds. Duke Math. J. 145(2008), 25–44. Google Scholar | DOI

[6] [6] Cox, B., Linowitz, B., McReynolds, D. B., and Miller, N., Local invariants and Galois cohomology. In preparation. Google Scholar

[7] [7] Garibaldi, S. and Saltman, D., Quaternion algebras with the same subfields. In: Quadratic forms, linear algebraic groups, and cohomology, Dev. Math. 18, Springer, New York, 2010, 225–238. Google Scholar

[8] [8] Komatsu, K., On the adèle rings of algebraic number fields. KodaiMath. Sem. Rep. 28(1976), 78–84. Google Scholar | DOI

[9] [9] Komatsu, K., On the adèle rings and zeta-functions of algebraic number fields. Kodai Math. J. 1(1978), 394–400. Google Scholar | DOI

[10] [10] Komatsu, K., On adèle rings of arithmetically equivalent fields. Acta Arith. 43(1984), 93–95. Google Scholar

[11] [11] Linowitz, B., Isospectral towers of Riemannian Manifolds. New York J. Math. 18(2012), 451–461. Google Scholar

[12] [12] Lubotzky, A., Samuels, B., and Vishne, U., Division Algebras and Non-Commensurable Isospectral Manifolds. Duke Math. J. 135(2006), 361–379. Google Scholar | DOI

[13] [13] Marcus, D. A., Number fields. Springer–Verlag, New York–Heidelberg, 1977. Google Scholar

[14] [14] McReynolds, D. B. and Reid, A.W., The genus spectrum of hyperbolic 3-manifolds. Math. Res. Lett.,to appear. Google Scholar

[15] [15] Meyer, J., Division algebras with infinite genus. Preprint. Google Scholar

[16] [16] Perlis, R., On the equation ζK(s) = ζK’ (s). J. Number Theory 9(1977), 342–360. Google Scholar | DOI

[17] [17] Pierce, R. S., Associative algebras. Grad. Texts in Math. 88, Springer-Verlag, New York–Berlin, 1982. Google Scholar

[18] [18] Platonov, V. and Rapinchuk, A., Algebraic groups and number theory. Pure Appl. Math. 139, Academic Press, Boston, MA, 1994. Google Scholar

[19] [19] Prasad, G. and Rapinchuk, A., Weakly commensurable arithmetic groups and isospectral locally symmetric spaces. Publ. Math. Inst. Hautes Étud. Sci. 109(2009), 113–184. Google Scholar

[20] [20] Reid, A. W., Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds. Duke Math. J. 65(1992), 215–228. Google Scholar | DOI

[21] [21] Serre, J.-P., Local fields. Grad. Texts in Math. 67, Springer-Verlag, New York–Berlin, 1979. Google Scholar

[22] [22] Witte-Morris, D., Introduction to arithmetic groups. http://people.uleth.ca/_dave.morris/books/IntroArithGroups.pdf Google Scholar

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