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Benjamin Linowitz; D. B. McReynolds; Nicholas Miller
Locally Equivalent Correspondences
(Correspondances Localement Équivalentes)
Annales de l'institut Fourier, 67 no. 2 (2017), p. 451-482, doi: 10.5802/aif.3088
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Class. Math.: 11R52, 11S25, 11E72, 16K50
Keywords: arithemetic equivalence, Brauer groups, Galois cohomology, maximal orders

Résumé - Abstract

Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois cohomology sets, and commensurability classes of arithmetic lattices in simple, inner algebraic groups. We show that under certain conditions, lattices corresponding to one another under our bijections have the same covolume and pro-congruence completion. We also make effective a finiteness result of Prasad and Rapinchuk.


[1] Menny Aka, “Arithmetic groups with isomorphic finite quotients”, J. Algebra 352 (2012) no. 1, p. 322-340 Article
[2] Grégory Berhuy, An introduction to Galois cohomology and its applications, London Mathematical Society Lecture Note Series 377, Cambridge University Press, 2010
[3] Armand Borel, “Commensurability classes and volumes of hyperbolic $3$-manifolds”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8 (1981), p. 1-33
[4] Armand Borel & Harish-Chandra, “Arithmetic subgroups of algebraic groups”, Ann. of Math. 75 (1962), p. 485-535 Article
[5] Armand Borel & Gopal Prasad, “Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups”, Inst. Hautes Études Sci. Publ. Math. 69 (1989), p. 119-171 Article
[6] Wieb Bosma, John Cannon & Catherine Playoust, “The Magma algebra system. I. The user language”, J. Symbolic Comput. 24 (1997) no. 3-4, p. 235-265 Article
[7] Ted Chinburg & Eduardo Friedman, “The smallest arithmetic hyperbolic three-orbifold”, Invent. Math. 86 (1986), p. 507-527 Article
[8] Ted Chinburg & Eduardo Friedman, “An embedding theorem for quaternion algebras”, J. London Math. Soc. 60 (1999), p. 33-44 Article
[9] Ted Chinburg, Emily Hamilton, Darren D. Long & Alan W. Reid, “Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds”, Duke Math. J. 145 (2008) no. 1, p. 25-44 Article
[10] Kenkichi Iwasawa, “On the rings of valuation vectors”, Ann. of Math. 57 (1953), p. 331-356 Article
[11] Keiichi Komatsu, “On adele rings of arithmetically equivalent fields”, Acta Arith. 43 (1984), p. 93-95
[12] Serge Lang, Algebraic number theory, Graduate Texts in Mathematics 110, Springer-Verlag, 1994
[13] Benjamin Linowitz, D. B. McReynolds, Paul Pollack & Lola Thompson, “Counting and effective rigidity in algebra and geometry”, http://arxiv.org/abs/1407.2294, 2014
[14] A. Lubotzky, B. Samuels & U. Vishne, “Division algebras and noncommensurable isospectral manifolds”, Duke Math. J. 135 (2006) no. 2, p. 361-397 Article
[15] Colin Maclachlan & Alan W. Reid, The Arithmetic of Hyperbolic 3–Manifolds, Graduate Texts in Mathematics 219, Springer-Verlag, 2003
[16] D. B. McReynolds, “Geometric Spectra and Commensurability”, Cand. Jour. Math. 67 (2015) no. 1, p. 184-197 Article |  MR 3292699
[17] D. B. McReynolds & Alan W. Reid, “The genus spectrum of a hyperbolic 3–manifold”, Math. Res. Lett. 21 (2014) no. 1, p. 169-185 Article
[18] Jürgen Neukirch, Alexander Schmidt & Kay Wingberg, Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften 323, Springer-Verlag, 2000
[19] Takashi Ono, “On algebraic groups and discontinuous groups”, Nagoya Math. J. 27 (1966), p. 279-322 Article
[20] Robert Perlis, “On the equation $\zeta _{K}(s)=\zeta _{K^{\prime }}(s)$”, J. Number Theory 9 (1977), p. 342-360 Article
[21] Richard S. Pierce, Associative algebras, Graduate Texts in Mathematics 88, Springer-Verlag, 1982
[22] Vladimir Platonov & Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics 139, Boston Academic Press, 1994
[23] Dipendra Prasad, “A refined notion of arithmetically equivalent number fields, and curves with isomorphic Jacobians”, http://arxiv.org/abs/1409.3173, 2014
[24] Gopal Prasad, “Volumes of S-Arithmetic Quotients of Semi-Simple Groups”, Inst. Hautes Études Sci. Publ. Math. 69 (1989), p. 91-117 Article
[25] Gopal Prasad & Andrei S. Rapinchuk, “Weakly commensurable arithmetic groups and isospectral locally symmetric spaces”, Publ. Math., Inst. Hautes Étud. Sci. 109 (2009), p. 113-1884 Article
[26] Alan W. Reid, “Isospectrality and commensurability of arithmetic hyperbolic $2$– and $3$–manifolds”, Duke Math. J. 65 (1992) no. 2, p. 215-228 Article
[27] Irving Reiner, Maximal orders, L.M.S. Monographs 5, London Academic Press, 1975  MR 393100
[28] Jean-Pierre Serre, Galois Cohomology, Lecture Notes in Mathematics 5, Springer-Verlag, 1994
[29] Bart de Smit & Robert Perlis, “Zeta functions do not determine class numbers”, Bull. Amer. Math. Soc. 31 (1994) no. 2, p. 213-215 Article
[30] Jacques Tits, “Reductive Groups over Local Fields”, Proc. Sympos. Pure Math. 33 (1979), p. 29-69 Article
[31] Dave Witte-Morris, “Introduction to Arithmetic Groups”, http://arxiv.org/abs/math/0106063, 2015