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Algebra Seminar
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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.