Local-global principles for embedding of fields with involution into simple algebras with involution

Details Local-global principles for embedding of fields with involution into simple algebras with involution Local-global principles for embedding of fields with involution into simple algebras with involution

2008-06-03

arxiv.org/abs/0806.0596v5

Mathematics

Number Theory

Scientific paper

In this paper we prove local-global principles for embedding of fields with involution into central simple algebras with involution over a global field. These should be of interest in study of classical groups over global fields. We deduce from our results that in a group of type D_n, n>4 even, two weakly commensurable Zariski-dense S-arithmetic subgroups are actually commensurable. A consequence of this result is that given an absolutely simple algebraic K-group G of type D_n, n>4 even, K a number field, any K-form G' of G having the same set of isomorphism classes of maximal K-tori as G, is necessarily K-isomorphic to G. These results lead to results about isolength and isospectral compact hyperbolic spaces of dimension 2n-1 with n even.

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