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Invariant fields of symplectic and orthogonal groups

✍ Scribed by David J. Saltman

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 271 KB
Volume: 258
Category: Article
ISSN: 0021-8693
DOI: 10.1016/s0021-8693(02)00640-3

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✦ Synopsis

The projective orthogonal and symplectic groups PO n (F ) and PSp n (F ) have a natural action on the F vector space

Here we assume F is an infinite field of characteristic not 2. If we assume there is more than one summand in V , then the invariant fields F (V ) PO n and F (V ) PSp n are natural objects. They are, for example, the centers of generic algebras with the appropriate kind of involution. This paper considers the rationality properties of these fields, in the case 1, 2, or 4 are the highest powers of 2 that divide n. We derive rationality when n is odd, or when 2 is the highest power, and stable rationality when 4 is the highest power. In a companion paper joint with Tignol, we prove retract rationality when 8 is the highest power of 2 dividing n. Back in this paper, along the way, we consider two generic ways of forcing a Brauer class to be in the image of restriction.

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