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

Andrei

Gopal Prasad

S. Rapinchuk

English

arXiv

In this paper we prove local–global principles for the existence of an embedding (E, σ) ↪ (A, τ) of a given global field E endowed with an involutive automorphism σ into a simple algebra A given with an involution τ in all situations except where A is a matrix algebra of even degree over a quaternion division algebra and τ is orthogonal (Theorem A of the introduction). Rather surprisingly, in the latter case we have a result which in some sense is opposite to the local–global principle, viz. algebras with involution locally isomorphic to (A, τ) are distinguished by their maximal subfields invariant under the involution (Theorem B of the introduction). These results can be used in the study of classical groups over global fields. In particular, we use Theorem B to complete the analysis of weakly commensurable Zariski-dense S-arithmetic groups in all absolutely simple algebraic groups of type different from D4 which was initiated in our paper [23]. More precisely, we prove that in a group of type Dn, n even > 4, two weakly commensurable Zariski-dense S-arithmetic subgroups are actually commensurable. As indicated in [23], this fact leads to results about length-commensurable and isospectral compact arithmetic hyperbolic manifolds of dimension 4n + 7, with n ≥ 1. The appendix contains a Galois-cohomological interpretation of our embedding theorems

Prasad, Gopal

Rapinchuk, Andrei S.

Indian Academy of Sciences

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

Publications of the IAS Fellows

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.311.5410

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

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Indian Academy of Sciences

Publications of the IAS Fellows