APPLICATIONS OF CLIFFORD ALGEBRAS TO INVOLUTIONS AND QUADRATIC FORMS

ABSTRACT

Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ϕ ∈ I ²(F) such that dim ϕ = 8, [C(ϕ)] = [D], and ϕ does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ _K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M _{2 ^m} (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 ⁿ⁻² − 2 and n ≥ 5, we show that q _σ ∈ I ⁿ (K), where q _σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.

Key Words:

• Involution
• Pfister form
• Quadratic form
• Reciprocity law

Mathematics Subject Classification:

• Primary 16W10, 11E81
• Secondary 16K20

ACKNOWLEDGMENTS

A preliminary version of this paper was made during my visit to the Université de Franche-Comté in Besancon, France. I express my thanks to Detlev Hoffmann for his hospitality. Also, I wish to thank Anne Quéguiner-Mathieu and Jean-Pierre Tignol for very fruitful discussions. The work on this publication was partially supported by INTAS 00-566, INTAS 99-00817, and SFB 478 “Geometrische Strukturen in der Mathematik.”

Notes

^#Communicated by R. Parimala.