### Mikhail Borovoi

Tel Aviv University

### R-equivalence on linear algebraic groups

**Friday 8th August, 12:05-12:55pm,
Carslaw 373. **
Let *X* be a smooth algebraic variety over a field *k*.
Manin in 1972 introduced the notion of R-equivalence on
*X*(*k*)
(the definition will be given in the talk). Let *X*(*k*)/R
denote the set of classes of R-equivalence. Colliot-Thelene
and Sansuc in 1977 computed the group *T*(*k*)/R, where
*T* is a *k*-torus.

Assume that *k* is a p-adic field, or a totally imaginary
number field, or *k*=*k*_0(*S*), where
*k*_0 is an algebraically closed
field of characteristic 0 and *S* is a *k*_0-surface.
It is known that for such
a field *k* we have *G*(*k*)/R=1 for any
simply connected semisimple group *G*.
Using this fact, we compute *G*(*k*)/R
for any connected linear *k*-group *G*,
and prove that the abelian group *G*(*k*)/R is a
*k*-birational invariant of *G*.

This is a joint work with B. Kunyavskii and P. Gille.