Manin obstruction: Difference between revisions
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In [[mathematics]], in the field of arithmetic algebraic geometry, the '''Manin obstruction''' 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 field]]s but not over a [[global field]]. | In [[mathematics]], in the field of arithmetic algebraic geometry, the '''Manin obstruction''' 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 field]]s but not over a [[global field]]. | ||
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* {{cite journal | author=Alexei N. Skorobogatov | title=Beyond the Manin obstruction | journal=Invent. Math. | volume=135 | issue=2 | pages=399-424 | year=1999 }} | * {{cite journal | author=Alexei N. Skorobogatov | title=Beyond the Manin obstruction | journal=Invent. Math. | volume=135 | issue=2 | pages=399-424 | year=1999 }} | ||
* {{cite book | title=Torsors and rational points | author=Alexei Skorobogatov | series=Cambridge Tracts in Mathematics | volume=144 | year=2001 | isbn=0521802377 | pages=1-7,112 }} | * {{cite book | title=Torsors and rational points | author=Alexei Skorobogatov | series=Cambridge Tracts in Mathematics | volume=144 | year=2001 | isbn=0521802377 | pages=1-7,112 }} | ||
Revision as of 15:21, 27 October 2008
In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction 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. 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, 250-258. ISBN 3-540-61223-8.
- Alexei N. Skorobogatov (1999). "Beyond the Manin obstruction". Invent. Math. 135 (2): 399-424.
- Alexei Skorobogatov (2001). Torsors and rational points, 1-7,112. ISBN 0521802377.