Locally Equivalent Correspondences
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, p. 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.
Received : 2015-05-26
Revised : 2016-02-18
Accepted : 2016-03-24
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3088
Classification:  11R52,  11S25,  11E72,  16K50
Keywords: arithemetic equivalence, Brauer groups, Galois cohomology, maximal orders
     author = {Linowitz, Benjamin and McReynolds, D. B. and Miller, Nicholas},
     title = {Locally Equivalent Correspondences},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {2},
     year = {2017},
     pages = {451-482},
     doi = {10.5802/aif.3088},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/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/item/AIF_2017__67_2_451_0/

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