# Commentarii Mathematici Helvetici

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**Volume 85, Issue 3, 2010, pp. 583–645**

**DOI: 10.4171/CMH/206**

Published online: 2010-05-23

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

Gopal Prasad^{[1]}and Andrei S. Rapinchuk

^{[2]}(1) University of Michigan, Ann Arbor, United States

(2) University of Virginia, Charlottesville, USA

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 `D`_{4} which was
initiated in our paper [23]. More precisely, we prove that in
a group of type `D`_{n}, `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 4`n` + 7, with `n` ≥ 1.
The appendix contains a Galois-cohomological interpretation of our
embedding theorems.

*Keywords: *Local–global principles, central simple algebras, involutions, arithmetic groups, locally symmetric spaces

Prasad Gopal, Rapinchuk Andrei: Local–global principles for embedding of fields with involution into simple algebras with involution. *Comment. Math. Helv.* 85 (2010), 583-645. doi: 10.4171/CMH/206