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Manin obstruction

From Encyclopedia of Mathematics
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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

How to Cite This Entry:
Brauer–Manin obstruction. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Brauer%E2%80%93Manin_obstruction&oldid=37488