Patching And Weak Approximation In Isometry Groups

Let R be a semilocal principal ideal domain. Two algebraic objects over R in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all completions of R and its fraction field. We prove that the number of isomorphism classes in the genus of unimodular quadratic spaces over ( not necessarily commutative) R-orders is always a finite power of 2, and under further assumptions, e.g., that the order is hereditary, this number is 1. The same result is also shown for related objects, e.g., systems of sesquilinear forms. A key ingredient in the proof is a weak approximation theorem for groups of isometries, which is valid over any (topological) base field, and even over semilocal base rings.

Rationally isomorphic hermitian forms and torsors of some non-reductive groups

Eva Bayer Fluckiger

Let R be a semilocal Dedekind domain. Under certain assumptions, we show that two (not necessarily unimodular) hermitian forms over an R-algebra with involution, which are rationally isomorphic and have isomorphic semisimple coradicals, are in fact isomorphic. The same result is also obtained for quadratic forms equipped with an action of a finite group. The results have cohomological restatements that resemble the Grothendieck-Serre conjecture, except the group schemes involved are not reductive. We show that these group schemes are closely related to group schemes arising in Bruhat-Tits theory. (C) 2017 Elsevier Inc. All rights reserved.

Academic Press Inc Elsevier Science2017

Orders that are étale-locally isomorphic

Eva Bayer Fluckiger, Mathieu Huruguen

Let R be a semilocal Dedekind domain with fraction field F. It is shown that two hereditary R-orders in central simple F-algebras that become isomorphic after tensoring with F and with some faithfully flat étale R-algebra are isomorphic. On the other hand, this fails for hereditary orders with involution. The latter stands in contrast to a result of the first two authors, who proved this statement for Hermitian forms over hereditary R-orders with involution. The results can be restated by means of étale cohomology and can be viewed as variations of the Grothendieck–Serre conjecture on principal homogeneous spaces of reductive group schemes. The relationship with Bruhat–Tits theory is also discussed.

2019

Anneau de Witt des G-formes et multiples de G-formes trace

Philippe Chabloz

This thesis deals with the study of G-forms and particulary the trace form of a G-Galois algebra. Let k be a field of characteristic not two. Let G be a finite group and L a G-Galois algebra over k. We define the trace form qL by qL(x, y) = TrL/k(xy) for all x, y in L. This is a bilinear symmetric form which is G-invariant. In other words, qL is a G-form. We know that L has a self-dual normal basis if and only if the trace form qL is G-isomorphic to the unit G-form q0. This is an important reason to classify the trace forms. This work contains two different parts. In the first part, we study G-forms in general, putting a ring structure on their Witt group. We then proved an analogue of Pfister's theorem - i.e. there is no zero divisor of odd dimension - when k[G] is semi-simple, k is big enough and G is abelian. Counter-examples are given when these conditions are not fulfilled. In the second part of this thesis, we study the trace form qL of a G-Galois algebra. E. Bayer-Fluckiger and H. W. Lenstra proved that if G is of odd order, then qL is always G-isomorphic to the unit form. If G is of even order, this is no longer the case. However, if the field k is of cohomological 2-dimension less than or equal to 1, then E. Bayer-Fluckiger and J.-P. Serre gave a necessary and sufficient condition - in terms of cohomological invariants - for the trace form qL to be isomorphic to the unit form. M. Monsurrò generalized this result to fields of virtual cohomological 2-dimension equal to 1. However, in higher cohomological dimensions, it becomes very difficult to classify the trace form itself. But it is possible to give general results if we consider multiples of the trace form or more generally the product of the trace form by a quadratic form. E. Bayer-Fluckiger formulated 2 conjectures about the possibility of finding a complete system of invariants for such a product when the quadratic form lies in a certain ideal of the Witt ring of k depending on the cohomological dimension of the field. In this work, we prove the first conjecture for all cohomological dimensions and the second one for a field of cohomological 2-dimension equal to 2. A more general conjecture is proved including the fields of virtual cohomological 2-dimension equal to 2. Finally, the second conjecture of E. Bayer-Fluckiger is proved in all cohomological dimensions, but only when either the characteristic of k is non zero or the group G is abelian or a 2-group, or k is big enough.

EPFL2004

Anneau de Witt des G-formes et multiples de G-formes trace

Philippe Chabloz

EPFL2004