In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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. The Manin obstruction is sometimes called the Brauer–Manin obstruction, as Manin used the
Brauer group of X to define it.
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
Alexei Skorobogatov, of varieties with trivial Manin obstruction which have points everywhere locally and yet no global points.
References
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Diophantine geometry
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