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 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. 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.