Abstract
We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise the symplectic result was shown in Becher (Invent Math 173(1):1–6, 2008) and the unitary result was partly shown in Black and Quéguiner-Mathieu (Enseign Math 60(3–4):377–395, 2014).
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Albert, A.: Structure of Algebras, American Mathematical Society Colloquium Publications 24. American Mathematical Society, New York (1968)
Bayer-Fluckiger, E., Parimala, R., Quéguiner-Mathieu, A.: Pfister involutions. Proc. Indian Acad. Sci. Math. Sci. 113, 365–377 (2003)
Becher, K.J.: A proof of the Pfister factor conjecture. Invent. Math. 173(1), 1–6 (2008)
Becher, K.J., Dolphin, A.: Non-hyperbolic splitting of quadratic pairs. J. Algebra Appl. 14(10), 1550138 (2015)
Becher, K.J., Dolphin, A.: Totally decomposable quadratic pairs. Math. Zeit. 284(1), 117–129 (2016)
Black, J., Quéguiner-Mathieu, A.: Involutions, odd-degree extensions and generic splitting. Enseign. Math. 60(3–4), 377–395 (2014)
Dolphin, A.: Orthogonal Pfister involutions in characteristic two. J. Pure Appl. Algebra 218(10), 1900–1915 (2014)
Dolphin, A., Quéguiner-Mathieu, A.: Symplectic involutions, quadratic pairs and function fields of conics. Preprint. https://www.math.uni-bielefeld.de/LAG/man/566.html (2016)
Elman, R., Karpenko, N., Merkurjev, A.: The Algebraic and Geometric Theory Quadratic Forms, American Mathematical Society Colloquium Publications, vol. 56. American Mathematical Society, Providence (2008)
Jacobson, N.: A note on hermitian forms. Bull. Am. Math. Soc. 46, 264–268 (1940)
Karpenko, N.A.: Hyperbolicity of orthogonal involutions. With an appendix by Jean-Pierre Tignol. Doc. Math. Extra volume: Andrei A. Suslin sixtieth birthday 371–392 (2010)
Knus, M.A.: Quadratic and Hermitian Forms over Rings, Grundlehren der mathematischen Wissenschaften, vol. 294. Springer, Berlin (1991)
Knus, M.A., Merkurjev, A.S., Rost, M., Tignol, J.-P.: The Book of Involutions, American Mathematical Society Colloquium Publications, vol. 44. American Mathematical Society, Providence (1998)
Pierce, R.: Associative Algebras. Graduate Texts in Mathematics. Springer, Berlin (1982)
Sah, C.-H.: A note on hermitian forms over fields of characteristic 2. Am. J. Math. 86(2), 262–270 (1964)
Scharlau, W.: Quadratic and Hermitian Forms, Volume 270 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1985)
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This work was supported by the DFG (The Pfister Factor Conjecture in Characteristic Two, BE 2614/4) and the FWO Odysseus programme (Explicit Methods in Quadratic Form Theory).
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Dolphin, A. Totally decomposable symplectic and unitary involutions. manuscripta math. 153, 523–534 (2017). https://doi.org/10.1007/s00229-016-0891-6
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DOI: https://doi.org/10.1007/s00229-016-0891-6