Embeddings of maximal tori in classical groups and explicit Brauer-Manin obstruction. (English) Zbl 1452.11037
In their paper [Comment. Math. Helv. 85, No. 3, 583–645 (2010; Zbl 1223.11047)] G. Prasad and A. S. Rapinchuk proved a Hasse principle for the existence of an embedding of a global field \(E\) with an involutive automorphism into a simple algebra \(A\) with a given involution \(\tau\) when \(\tau\) is symplectic or when \(\tau\) is orthogonal but \(A \neq M_{2n}(D)\) for a quaternion division algebra.
The first author of this paper had obtained combinatorial criteria for Hasse principle to hold when \(A\) is a matrix algebra and \(\tau\) is orthogonal [J. Eur. Math. Soc. (JEMS) 17, No. 7, 1629–1656 (2015; Zbl 1326.11008)].
Also, building on results of M. Borovoi [Math. Ann. 314, No. 3, 491–504 (1999; Zbl 0966.14017)], the second author proved that the Brauer-Manin obstruction is the only obstruction to Hasse principle holding [Comment. Math. Helv. 89, No. 3, 671–717 (2014; Zbl 1321.11043)].
These different points of view are explained by a construction of obstruction to the Hasse principle proved in the paper under review. In particular, the authors define the notion of an ‘oriented embedding’ of a field with involution into a central simple algebra with involution. Using this, they extend the main result of Prasad-Rapinchuk to show that existence of oriented embeddings locally implies a global embedding.
More generally, when \(E\) is an étale algebra with involution \(\sigma\) over a global field \(K\), the authors define a group \(Ш(E,\sigma)\) closely connected to a Tate-Shafarevich group; this group \(Ш(E,\sigma)\) encodes ramification properties of the components of \((E, \sigma)\). Associated to oriented embeddings of \((E, \sigma)\) into \((A. \tau)\) locally, the authors define local embedding data which enables them to obtain a homomorphism \(f\) form \(Ш(E, \sigma)\) to \(\mathbb{Z}/2 \mathbb{Z}\).
The authors’ necessary and sufficient criterion for the Hasse principle asserts that given oriented embeddings of \((E, \sigma)\) into \((A, \tau)\) over all completions \(K_v\), there is a global embedding over \(K\) if, and only if, the corresponding \(f : Ш(E, \sigma) \rightarrow \mathbb{Z}/2 \mathbb{Z}\) is the zero map.
The first author of this paper had obtained combinatorial criteria for Hasse principle to hold when \(A\) is a matrix algebra and \(\tau\) is orthogonal [J. Eur. Math. Soc. (JEMS) 17, No. 7, 1629–1656 (2015; Zbl 1326.11008)].
Also, building on results of M. Borovoi [Math. Ann. 314, No. 3, 491–504 (1999; Zbl 0966.14017)], the second author proved that the Brauer-Manin obstruction is the only obstruction to Hasse principle holding [Comment. Math. Helv. 89, No. 3, 671–717 (2014; Zbl 1321.11043)].
These different points of view are explained by a construction of obstruction to the Hasse principle proved in the paper under review. In particular, the authors define the notion of an ‘oriented embedding’ of a field with involution into a central simple algebra with involution. Using this, they extend the main result of Prasad-Rapinchuk to show that existence of oriented embeddings locally implies a global embedding.
More generally, when \(E\) is an étale algebra with involution \(\sigma\) over a global field \(K\), the authors define a group \(Ш(E,\sigma)\) closely connected to a Tate-Shafarevich group; this group \(Ш(E,\sigma)\) encodes ramification properties of the components of \((E, \sigma)\). Associated to oriented embeddings of \((E, \sigma)\) into \((A. \tau)\) locally, the authors define local embedding data which enables them to obtain a homomorphism \(f\) form \(Ш(E, \sigma)\) to \(\mathbb{Z}/2 \mathbb{Z}\).
The authors’ necessary and sufficient criterion for the Hasse principle asserts that given oriented embeddings of \((E, \sigma)\) into \((A, \tau)\) over all completions \(K_v\), there is a global embedding over \(K\) if, and only if, the corresponding \(f : Ш(E, \sigma) \rightarrow \mathbb{Z}/2 \mathbb{Z}\) is the zero map.
Reviewer: Balasubramanian Sury (Bangalore)
MSC:
| 11E57 | Classical groups |
| 11E88 | Quadratic spaces; Clifford algebras |
| 20G30 | Linear algebraic groups over global fields and their integers |
Keywords:
Hasse principle; algebras with Involutions; eEmbedding problem; Brauer-Manin construction; Tate-Shafarevich groupReferences:
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