Manin obstruction

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