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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), 189–200.

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 Mathématiciens (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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  • $\begingroup$ For reference: I. Manin, "Le groupe de Brauer-Grothendieck en géométrie diophantienne", Proc. Internat. Congr. Math. (Nice), 1970. $\endgroup$
    – user19475
    Commented Jan 10, 2010 at 12:49
  • $\begingroup$ You might be especially interested in Theorème 4.1 in Harari's article. $\endgroup$
    – Chandan Singh Dalawat
    Commented Jan 10, 2010 at 12:57
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    $\begingroup$ 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. $\endgroup$
    – Chandan Singh Dalawat
    Commented Jan 10, 2010 at 13:26
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    $\begingroup$ @Chandan This has not been proved for curves, only for torsors of abelian varieties. $\endgroup$
    – Felipe Voloch
    Commented Jan 10, 2010 at 15:13
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    $\begingroup$ @Felipe : You are right. It is perhaps expected to be true for curves of genus >1. $\endgroup$
    – Chandan Singh Dalawat
    Commented Jan 11, 2010 at 4:13

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