An invariant for symplectic involutions on central simple algebras of degree divisible by 4 over fields of characteristic different from 2 is defined on the basis of Rost's cohomological invariant of degree 3 for torsors under symplectic groups. We relate this invariant to trace forms and show how its triviality yields a decomposability criterion for algebras of degree 8 with symplectic involution.
Abstract. Using a cohomological obstruction, we construct examples of absolutely simple adjoint classical groups of type 2 An with n ≡ 3 mod 4, Cn or 1 Dn with n ≡ 0 mod 4, which are not R-trivial hence not stably rational.
Let ℓ be a prime and let L/Q be a Galois number field with Galois group isomorphic to Z/ℓZ. We show that the shape of L, see definition 1.2, is either 1 2 A ℓ−1 or a fixed sub lattice depending only on ℓ; such a dichotomy in the value of the shape only depends on the type of ramification of L. This work is motivated by a result of Bhargava and Shnidman, and a previous work of the first named author, on the shape of Z/3Z number fields.
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