Manin obstruction

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. 
• Alexei N. Skorobogatov (1999). "Beyond the Manin obstruction". Invent. Math. 135 (2): 399–424. doi:10.1007/s002220050291. 
• Alexei Skorobogatov (2001). Torsors and rational points. Cambridge Tracts in Mathematics. 144. Cambridge: Cambridge Univ. Press. pp. 1–7,112. ISBN 0521802377. 

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