Difference between revisions of "Manin obstruction"
From Encyclopedia of Mathematics
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==References== | ==References== | ||
| − | * Serge Lang ''Survey of Diophantine geometry'' Springer-Verlag (1997) ISBN 3-540-61223-8 | + | * 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 "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. | + | * 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. |
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Latest revision as of 19:39, 17 November 2023
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.
How to Cite This Entry:
Manin obstruction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Manin_obstruction&oldid=54515
Manin obstruction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Manin_obstruction&oldid=54515