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.
- Serge Lang (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 250–258. ISBN 3-540-61223-8. Zbl 0869.11051.
- Alexei N. Skorobogatov (1999). "Beyond the Manin obstruction". Appendix A by S. Siksek: 4-descent. 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.
|This number theory-related article is a stub. You can help Wikipedia by expanding it.|
|This geometry-related article is a stub. You can help Wikipedia by expanding it.|