Résumé des résultats (Abstract)
(Anglais)
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Our research has followed two main directions :
1) Multiples of trace forms over fields of virtual cohomological dimension 1 and 2
Let K be a field of characteristic ¹ 2, let G be a finite group, and let L be a G-Galois algebra over K. Let qL be the associated trace form. For any positive integer m, we denote by [m]qL the orthogonal direct sum of m copies of qL.
The aim of this research is to find criteria for the G-isomorphism of [m]qL and [m]qL' , where L and L' are two G-Galois algebras. We show that if m is odd, then [m]qL @ [m]qL' if and only if qL @ qL' . Therefore, it is enough to solve the above problem when m is a power of 2. We do this when m = 2d, with d greater than or equal to the virtual cohomological dimension of the field. The next step is to give a cohomological criterion for the [m]qL @ qL 'to hold when m = 2 d-1, d being the virtual cohomological dimension of K. We complete this step in the case d=2.
[1] E. Bayer-Fluckiger and Ph. Chabloz, Multiples of G-forms, in preparation
[2] E. Bayer-Fluckiger, M. Monsurro, R. Parimala and R. Schoof, Multiples of trace forms over fields of virtual cohomological dimension 1 and 2, to appear
2) Cohomological invariants of algebras with involution
and of linear algebraic groups
Let K be a field of characteristic ¹ 2, and let A be a central simple F-algebra. Let n be the degree of A. Defining cohomological invariants for involutions of various types of A is an important and active research topic. Several results exist already in the case of orthogonal and unitary involutions. The aim of the present research was to define invariants also in the case of symplectic involutions.
Let s : A ® A be a symplectic involution of the algebra A. We define a cohomological invariant with values in H3(F, m2) , called discriminant, and denoted by D(s). We show that the triviality of D(s) is related to the decomposability of s in a tensor product of 3 involutions when deg (A) = 8. We then use this invariant to construct examples of absolutely simple adjoint classical groups that are not R-trivial, hence not stably rational.
[1] G. Berhuy, M. Monsurro and J.-P. Tignol, The discriminant of a symplectic involution, Pacific Journal of Math., to appear
[2] G. Berhuy, M. Monsurro and J.-P. Tignol, Cohomological invariants and R-equivalence of adjoint classical groups, to appear
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