Raman Parimala

Universal Unramified Cohomology of Cubic Fourfolds Containing a Plane

We prove the universal triviality of the third unramified cohomology group of a very general complex cubic fourfold containing a plane. The proof uses results on the unramified cohomology of quadrics due to Kahn, Rost, and Sujatha.

Quadratic spaces over Laurent extensions of Dedekind domains

Let Abea Dedekind domain in which 2 is invertible. We show in this paper that any isotropic quadratic space over R(T, 71) is isometric to n ii q — q ± HiP), with rank P > n, where if(P) denotes the hyperbolic space. 1. Some preliminary results. In this section we prove some lemmas which will be used in the later sections. We assume throughout the paper that in all the rings considered, 2 is invertible. If a,, q2 are quadratic spaces over R, a, J. Tq2 denotes the quadratic space (a, ®R R(T, T~x)) _L T(a2 ®R R(T, T~x)) over R(T, T~x).

Hasse principle for G-trace forms

Let be a global field of characteristic not 2. We prove a local-global principle for the existence of self-dual normal bases, and more generally for the isomorphism of -trace forms, for -Galois algebras over .

Embeddings of maximal tori in classical groups over local and global fields

Embeddings of maximal tori in classical groups over fields of characteristic not 2 are the subject matter of several recent papers. The aim of the present paper is to give necessary and sufficient conditions for such an embedding to exist, when the base field is a local field, or the field of real numbers. This completes the results of [3], where a complete criterion is given for the Hasse principle to hold when the base field is a global field.

R-EQUIVALENCE IN ADJOINT CLASSICAL GROUPS OVER FIELDS OF VIRTUAL COHOMOLOGICAL DIMENSION 2

Let F be a field of characteristic not 2 whose virtual cohomological dimension is at most 2. Let G be a semisimple group of adjoint type defined over F. Let RG(F) denote the normal subgroup of G(F) consisting of elements R-equivalent to identity. We show that if G is of classical type not containing a factor of type D n , G(F)/RG(F) = 0. If G is a simple classical adjoint group of type D n , we show that if F and its multi-quadratic extensions satisfy strong approximation property, then G(F)/RG(F) = 0. This leads to a new proof of the R-triviality of F-rational points of adjoint classical groups defined over number fields.