On the $R$-equivalence on algebraic tori over pseudoglobal fields

V.I.Andriychuk

Let $T/K$ be an algebraic torus defined over a pseudoglobal field, i.e. over an algebraic function field in one variable with pseudofinite constant field. We show that the groups of $R$–equivalence and of $ \mathrm{Br}$–equivalence for $T(k)$ are finite. Moreover, the connections of these groups with the Tate-Shafarevich groups and with the defects of weak approximation are analogous to the case of a global ground field.

1. Andriychuk V. On the algebraic tori over some function fields, Mat. Studii 12 (1999), no.2, 115--126.

2. Ax J. The elementary theory of finite fields, Ann. of Math. 88, (1968), no.2, 239--271.

3. Colliot-Théléne J.-L. et Sansuc J.-J. La $R$-équivalence sur les tores, Ann. Sc.E.N.S. t.10 (1976), 175--230.

4. Colliot-Théléne J.-L. et Sansuc J.-J. La descente sur les varietétes rationnelles, II, Duke Math. J. 54 (1987), 375--492.

5. Efrat I. A Hasse principle for function fields over PAC fields, Israel J. Math. 122 (2001), 43--60.

6. Fried M. and Jarden M. Field Arithmetic, Springer, Heidelberg, 1986.

7. Gi Gille P. Spécialisation de la $R$-équivalence pour les groupes réductifs, Preprint.

8. Manin Yu.I. Le groupe de Brauer-Grothendieck en géometrie diophantienne, Actes du Congrés intern. math. Nice 1 (1970), 401--411.

9. Manin Yu.I. Cubic forms : Algebra, Geometry, Arithmetic, North Holland, 1986, 2nd ed.

10. Milne J.S. Arithmetic Duality Theorems, Academic Press, Inc. Boston, 1986.

11. Rim D.S. and Whaples G. Global norm-residue map over quasi-finite fields, Nagoya Math. J, 27 (1968), no.1, 323--329.

12. Sansuc J.-J. Groupes de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J.reine. angew.Math. Bd. 327 (1981), 12--80.

13. Serre J.-P. Corps locaux, Paris, 1962.

14. Воскресенский В.Е. Алгебраические торы.--М.: Наука, 1977, 222 с.

	On the $R$-equivalence on algebraic tori over pseudoglobal fields
Author	V.I.Andriychuk v_andriychuk@mail.ru, topos@franko.lviv.ua Department of Mathematics and Mechanics, Ivan Franko Lviv National University
Abstract	Let $T/K$ be an algebraic torus defined over a pseudoglobal field, i.e. over an algebraic function field in one variable with pseudofinite constant field. We show that the groups of $R$–equivalence and of $ \mathrm{Br}$–equivalence for $T(k)$ are finite. Moreover, the connections of these groups with the Tate-Shafarevich groups and with the defects of weak approximation are analogous to the case of a global ground field.
Keywords	algebraic torus, pseudoglobal field, Tate-Shafarevich group
DOI	doi:10.30970/ms.22.2.176-183
Reference	1. Andriychuk V. On the algebraic tori over some function fields, Mat. Studii 12 (1999), no.2, 115--126. 2. Ax J. The elementary theory of finite fields, Ann. of Math. 88, (1968), no.2, 239--271. 3. Colliot-Théléne J.-L. et Sansuc J.-J. La $R$-équivalence sur les tores, Ann. Sc.E.N.S. t.10 (1976), 175--230. 4. Colliot-Théléne J.-L. et Sansuc J.-J. La descente sur les varietétes rationnelles, II, Duke Math. J. 54 (1987), 375--492. 5. Efrat I. A Hasse principle for function fields over PAC fields, Israel J. Math. 122 (2001), 43--60. 6. Fried M. and Jarden M. Field Arithmetic, Springer, Heidelberg, 1986. 7. Gi Gille P. Spécialisation de la $R$-équivalence pour les groupes réductifs, Preprint. 8. Manin Yu.I. Le groupe de Brauer-Grothendieck en géometrie diophantienne, Actes du Congrés intern. math. Nice 1 (1970), 401--411. 9. Manin Yu.I. Cubic forms : Algebra, Geometry, Arithmetic, North Holland, 1986, 2nd ed. 10. Milne J.S. Arithmetic Duality Theorems, Academic Press, Inc. Boston, 1986. 11. Rim D.S. and Whaples G. Global norm-residue map over quasi-finite fields, Nagoya Math. J, 27 (1968), no.1, 323--329. 12. Sansuc J.-J. Groupes de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J.reine. angew.Math. Bd. 327 (1981), 12--80. 13. Serre J.-P. Corps locaux, Paris, 1962. 14. Воскресенский В.Е. Алгебраические торы.--М.: Наука, 1977, 222 с.
Pages	176-183
Volume	22
Issue	2
Year	2004
Journal	Matematychni Studii
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