View source for Manin obstruction

''Brauer–Manin obstruction''

An invariant attached to a geometric object $X$ which measures the failure of the [[Hasse principle]] for $X$: that is, if the obstruction is non-trivial, then $X$ may have points over all [[local field]]s but not over a [[global field]].

For [[abelian variety|abelian varieties]] the Manin obstruction is just the [[Tate-Shafarevich group]] and fully accounts for the failure of the local-to-global principle.  There are however examples, due to Skorobogatov, of varieties with trivial Manin obstruction which have points everywhere locally and yet no global points. 

==References==
* Serge Lang ''Survey of Diophantine geometry'' Springer-Verlag (1997) ISBN 3-540-61223-8. {{ZBL|0869.11051}}.  pp.250–258. 
* Alexei Skorobogatov  "Beyond the Manin obstruction" (with Appendix A by S. Siksek: 4-descent). ''Inventiones Mathematicae'' '''135''' no.2 (1999) 399–424. {{DOI|10.1007/s002220050291}}.  {{ZBL|0951.14013}}. 
* Alexei Skorobogatov ''Torsors and rational points'' Cambridge Tracts in Mathematics '''144''' Cambridge University Press (2001) ISBN 0-521-80237-7 {{ZBL|0972.14015}}.  pp.1–7,112.

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