ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 12,14,18 Voskresenskii, Valentin E., Samara State University, Russia Stably rational algebraic tori Let X be an algebraic variety over a field k. We call X stably rational if X×_{k}A^{m} is rational for some m. In the category of algebraic tori we have the criterion of stable rationality which enables us to construct any stably rational tori [1]. However, among them one has’t yet found a nonrational torus. Conjecture. Any stably rational torus over a field k is rational over k. We suggest a new approach to this conjecture. In particular, we prove the rationality of any stably rational torus with a cyclic splitting field. Reference: [1] Voskresenskii V. E. The geometry of linear algebraic groups, Proc. Steklov Inst. Math., 132, 1973, 173-183.