Locally Equivalent Correspondences
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 451-482.

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.

Étant donnée une paire de corps de nombres dont les anneaux d’ adèles sont isomorphes, nous construisons des bijections entre certains objets associés à la paire. Par exemple, nous construisons un isomorphisme de groupes de Brauer qui commute avec la restriction. Nous construisons en outre des bijections entre algébres centrales simples, ordres maximaux, ensembles de cohomologie galoisienne, et classes de commensurabilité de réseaux arithmétiques dans des formes intérieures de groupes algébriques simples. Nous montrons que, sous des hypothèses convenables, des réseaux se correspondant l’un à l’autre sous nos bijections ont le même covolume et la même complétion pro-congruence. Nous rendons aussi effectif un rèsultat de finitude de Prasad et Rapinchuk.

Published online:
DOI: 10.5802/aif.3088
Classification: 11R52, 11S25, 11E72, 16K50
Keywords: arithemetic equivalence, Brauer groups, Galois cohomology, maximal orders
Mot clés : équivalence arithmétique, groupes de Brauer, cohomologie galoisienne, ordres maximaux
Linowitz, Benjamin 1; McReynolds, D. B. 2; Miller, Nicholas 2

1 Department of Mathematics University of Michigan Ann Arbor, MI 48109 (USA)
2 Department of Mathematics Purdue University West Lafayette, IN 47907 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Linowitz, Benjamin and McReynolds, D. B. and Miller, Nicholas},
     title = {Locally {Equivalent} {Correspondences}},
     journal = {Annales de l'Institut Fourier},
     pages = {451--482},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {2},
     year = {2017},
     doi = {10.5802/aif.3088},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3088/}
AU  - Linowitz, Benjamin
AU  - McReynolds, D. B.
AU  - Miller, Nicholas
TI  - Locally Equivalent Correspondences
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 451
EP  - 482
VL  - 67
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3088/
DO  - 10.5802/aif.3088
LA  - en
ID  - AIF_2017__67_2_451_0
ER  - 
%0 Journal Article
%A Linowitz, Benjamin
%A McReynolds, D. B.
%A Miller, Nicholas
%T Locally Equivalent Correspondences
%J Annales de l'Institut Fourier
%D 2017
%P 451-482
%V 67
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3088/
%R 10.5802/aif.3088
%G en
%F AIF_2017__67_2_451_0
Linowitz, Benjamin; McReynolds, D. B.; Miller, Nicholas. Locally Equivalent Correspondences. Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 451-482. doi : 10.5802/aif.3088. https://aif.centre-mersenne.org/articles/10.5802/aif.3088/

[1] Aka, Menny Arithmetic groups with isomorphic finite quotients, J. Algebra, Volume 352 (2012) no. 1, pp. 322-340 | DOI

[2] Berhuy, Grégory An introduction to Galois cohomology and its applications, London Mathematical Society Lecture Note Series, 377, Cambridge University Press, 2010, xi+315 pages

[3] Borel, Armand Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 8 (1981), pp. 1-33

[4] Borel, Armand; Harish-Chandra Arithmetic subgroups of algebraic groups, Ann. of Math., Volume 75 (1962), pp. 485-535 | DOI

[5] Borel, Armand; Prasad, Gopal Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Inst. Hautes Études Sci. Publ. Math., Volume 69 (1989), pp. 119-171 | DOI

[6] Bosma, Wieb; Cannon, John; Playoust, Catherine The Magma algebra system. I. The user language, J. Symbolic Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | DOI

[7] Chinburg, Ted; Friedman, Eduardo The smallest arithmetic hyperbolic three-orbifold, Invent. Math., Volume 86 (1986), pp. 507-527 | DOI

[8] Chinburg, Ted; Friedman, Eduardo An embedding theorem for quaternion algebras, J. London Math. Soc., Volume 60 (1999), pp. 33-44 | DOI

[9] Chinburg, Ted; Hamilton, Emily; Long, Darren D.; Reid, Alan W. Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds, Duke Math. J., Volume 145 (2008) no. 1, pp. 25-44 | DOI

[10] Iwasawa, Kenkichi On the rings of valuation vectors, Ann. of Math., Volume 57 (1953), pp. 331-356 | DOI

[11] Komatsu, Keiichi On adele rings of arithmetically equivalent fields, Acta Arith., Volume 43 (1984), pp. 93-95

[12] Lang, Serge Algebraic number theory, Graduate Texts in Mathematics, 110, Springer-Verlag, 1994, xiii+357 pages

[13] Linowitz, Benjamin; McReynolds, D. B.; Pollack, Paul; Thompson, Lola Counting and effective rigidity in algebra and geometry (2014) (http://arxiv.org/abs/1407.2294)

[14] Lubotzky, A.; Samuels, B.; Vishne, U. Division algebras and noncommensurable isospectral manifolds, Duke Math. J., Volume 135 (2006) no. 2, pp. 361-397 | DOI

[15] Maclachlan, Colin; Reid, Alan W. The Arithmetic of Hyperbolic 3–Manifolds, Graduate Texts in Mathematics, 219, Springer-Verlag, 2003, xiii+463 pages

[16] McReynolds, D. B. Geometric Spectra and Commensurability, Cand. Jour. Math., Volume 67 (2015) no. 1, pp. 184-197 | DOI

[17] McReynolds, D. B.; Reid, Alan W. The genus spectrum of a hyperbolic 3–manifold, Math. Res. Lett., Volume 21 (2014) no. 1, pp. 169-185 | DOI

[18] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften, 323, Springer-Verlag, 2000, xv+699 pages

[19] Ono, Takashi On algebraic groups and discontinuous groups, Nagoya Math. J., Volume 27 (1966), pp. 279-322 | DOI

[20] Perlis, Robert On the equation ζ K (s)=ζ K ' (s), J. Number Theory, Volume 9 (1977), pp. 342-360 | DOI

[21] Pierce, Richard S. Associative algebras, Graduate Texts in Mathematics, 88, Springer-Verlag, 1982, xii+436 pages

[22] Platonov, Vladimir; Rapinchuk, Andrei Algebraic groups and number theory, Pure and Applied Mathematics, 139, Boston Academic Press, 1994, xi+614 pages

[23] Prasad, Dipendra A refined notion of arithmetically equivalent number fields, and curves with isomorphic Jacobians (2014) (http://arxiv.org/abs/1409.3173)

[24] Prasad, Gopal Volumes of S-Arithmetic Quotients of Semi-Simple Groups, Inst. Hautes Études Sci. Publ. Math., Volume 69 (1989), pp. 91-117 | DOI

[25] Prasad, Gopal; Rapinchuk, Andrei S. Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 109 (2009), pp. 113-1884 | DOI

[26] Reid, Alan W. Isospectrality and commensurability of arithmetic hyperbolic 2– and 3–manifolds, Duke Math. J., Volume 65 (1992) no. 2, pp. 215-228 | DOI

[27] Reiner, Irving Maximal orders, L.M.S. Monographs, 5, London Academic Press, 1975, xii+395 pages

[28] Serre, Jean-Pierre Galois Cohomology, Lecture Notes in Mathematics, 5, Springer-Verlag, 1994, ix+181 pages

[29] de Smit, Bart; Perlis, Robert Zeta functions do not determine class numbers, Bull. Amer. Math. Soc., Volume 31 (1994) no. 2, pp. 213-215 | DOI

[30] Tits, Jacques Reductive Groups over Local Fields, Proc. Sympos. Pure Math., Volume 33 (1979), pp. 29-69 | DOI

[31] Witte-Morris, Dave Introduction to Arithmetic Groups (2015) (http://arxiv.org/abs/math/0106063)

Cited by Sources: