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## Brauer-Manin obstruction and Tate-Shafarevich group of an Abelian variety

I read that the Brauer-Manin obstruction $A(\mathbb{A}_K)^{\mathbf{Br}}$ of an Abelian variety $A$ over a number field $K$ equals (naturally?) its Tate-Shafarevich group $\mathrm{III}(A)$.

Is this true? And if so, where can I find a proof?

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The quotient of what you called the Brauer-Manin obstruction by the closure of $A(K)$ within it is related to the divisible part of Sha. In particular, if Sha has no divisible part (e.g. if it is finite) then the Brauer-Manin obstruction is the closure of $A(K)$. See L. Wang, Brauer-Manin obstruction to weak approximation on abelian varieties, Israel J. Math. 94 (1996), 189â€“200.

Note that these two groups in your question are very different, so they can't be equal. For instance, Sha is torsion, but the Brauer-Manin obstruction usually isn't.

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Manin's ICM talk at Nice (1970) might be a good place to start.

If you read French, an elementary introduction to the local-to-global principle and to the Manin obstruction can be found on David Harari's homepage. This article also appeared in the Gazette des MathÃ©maticiens (Janvier 2006).

A detailed treatment can be found in the book

MR1845760 (2002d:14032) Skorobogatov (Alexei), Torsors and rational points. Cambridge Tracts in Mathematics, 144. Cambridge University Press, Cambridge, 2001. viii+187 pp.

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For reference: I. Manin, "Le groupe de Brauer-Grothendieck en géométrie diophantienne", Proc. Internat. Congr. Math. (Nice), 1970. – norondion Jan 10 2010 at 12:49
You might be especially interested in Theorème 4.1 in Harari's article. – Chandan Singh Dalawat Jan 10 2010 at 12:57
Strictly speaking, there is no obstruction whatsoever to the existence of rational point on an abelian variety : the origin is a rational point ! What you have in mind is that if C is smooth projective curve of genus >0 with jacobian J, or if C is a torsor under an abelian variety J, everything being defined over some number field K, and if C has K_v-points at every place v of K, then the Manin obstruction to the existence of a K-point is the only one as long as Sha(J) is finite. – Chandan Singh Dalawat Jan 10 2010 at 13:26
@Chandan This has not been proved for curves, only for torsors of abelian varieties. – Felipe Voloch Jan 10 2010 at 15:13
@Felipe : You are right. It is perhaps expected to be true for curves of genus >1. – Chandan Singh Dalawat Jan 11 2010 at 4:13
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