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In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction (named after Yuri Manin) is attached to a geometric object X which measures the failure of the Hasse principle for X: that is, if the value of the obstruction is non-trivial, then X may have points over all local fields but not over a global field.

For abelian varieties the Manin obstruction is just the Tate-Shafarevich group and fully accounts for the failure of the local-to-global principle (under the assumption that the Tate-Shafarevich group is finite). 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 (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 250–258. ISBN 3-540-61223-8. Zbl 0869.11051.
Alexei N. Skorobogatov (1999). Appendix A by S. Siksek: 4-descent. "Beyond the Manin obstruction". Inventiones Mathematicae 135 (2): 399–424. doi:10.1007/s002220050291. Zbl 0951.14013.
Alexei Skorobogatov (2001). Torsors and rational points. Cambridge Tracts in Mathematics 144. Cambridge: Cambridge University Press. pp. 1–7,112. ISBN 0-521-80237-7. Zbl 0972.14015.

Mathematics Encyclopedia