Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-08T02:33:06.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient evaluation of the Brauer–Manin obstruction

Published online by Cambridge University Press:  12 February 2007

MARTIN BRIGHT
MARTIN BRIGHT*
Affiliation:
Department of Mathematics, University of Liverpool, Peach Street, Liverpool L69 7ZL. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The Brauer–Manin obstruction is a concept which has been very effective in finding counter-examples to the Hasse principle, that is, sets of polynomial equations which have solutions in every completion of the rational numbers but have no rational solutions. The standard way of calculating the Brauer–Manin obstruction involves listing all the p-adic solutions to some accuracy, at finitely many primes p; this is a process which may be time-consuming. The result described in this paper shows that, at some primes, we do not need to list all p-adic solutions, but only those lying over a closed subset; and, at other primes, we need only to list solutions modulo p.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 1 , January 2007 , pp. 13 - 23
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bright, M. and SirSwinnerton–Dyer, Peter. Computing the Brauer–Manin obstructions. Math. Proc. Camb. Phil. Soc. 137 (1) (2004), 1–16.CrossRefGoogle Scholar
[2] Bright, M. J.. Brauer groups of diagonal quartic surfaces. J. Symbolic Comput. (2005), to appear.Google Scholar
[3] Colliot-Thélène, J.-L. and SirSwinnerton–Dyer, Peter. Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties. J. Reine Angew. Math. 453 (1994), 49–112.Google Scholar
[4] Colliot–Thélène, J.-L., Sansuc, J.-J. and Swinnerton–Dyer, P.. Intersections of two quadrics and Châtelet surfaces. I. J. Reine Angew. Math. 373 (1987), 37–107.Google Scholar
[5] Colliot–Thélène, J.-L., Sansuc, J.-J. and Swinnerton–Dyer, P.. Intersections of two quadrics and Châtelet surfaces. II. J. Reine Angew. Math. 374 (1987), 72–168.Google Scholar
[6] Colliot–Thélène, J.-L., Skorobogatov, A. N. and SirSwinnerton–Dyer, Peter. Hasse principle for pencils of curves of genus one whose jacobians have rational 2-division points. Invent. Math. 134 (1998), 579–650.Google Scholar
[7] Flynn, E. V.. The Hasse principle and the Brauer-Manin obstruction for curves. Manuscripta Math. 115 (4) (2004), 437–466.CrossRefGoogle Scholar
[8] Grothendieck, A.. Le groupe de Brauer I, II, III. In Giraud, J. et al. , editors, Dix Exposés sur la Cohomologie des Schémas. Adv. Stud. Math. 3 (1968), 46–188.Google Scholar
[9] Grothendieck, A.. Séminaire de Géometrie Algébrique du Bois-Marie SGA 1: Revêtements Étales et Groupe Fondamental. Lecture Notes in Mathematics No. 224 (Springer-Verlag, 1971).Google Scholar
[10] Harari, D.. Méthode des fibrations et obstruction de Manin. Duke Math. J. 75 (1) (1994), 221–260.CrossRefGoogle Scholar
[11] Hartshorne, R.. Algebraic Geometry. Graduate Texts in Mathematics No. 52 (Springer-Verlag, 1977).CrossRefGoogle Scholar
[12] Manin, Yu. I.. Cubic forms: Algebra, Geometry, Arithmetic (North-Holland Publishing Co., 1974). Translated from Russian by M. Hazewinkel, North-Holland Mathematical Library, Vol. 4.Google Scholar
[13] Manin, Yu. I.. Le groupe de Brauer–Grothendieck en géométrie diophantienne. In Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1 (Gauthier-Villars, 1971), 401–411.Google Scholar
[14] Milne, J. S.. Class field theory. Online notes, available on the author's web site at http://www.jmilne.org/math (1997).Google Scholar
[15] Serre, J.-P.. Corps Locaux, volume VIII of Publications de l'Institut de Mathématique de l'Université de Nancago (Hermann, 1968).Google Scholar
[16] SirSwinnerton–Dyer, Peter. Brauer–Manin obstructions on some Del Pezzo surfaces. Math. Proc. Camb. Phil. Soc. 125 (1999), 193–198.CrossRefGoogle Scholar
[17] SirSwinnerton–Dyer, Peter. Arithmetic of diagonal quartic surfaces, II. Proc. London Math. Soc. 80 (2000), 513–544.CrossRefGoogle Scholar