I read that the BrauerManin obstruction $A(\mathbb{A}_K)^{\mathbf{Br}}$ of an Abelian variety $A$ over a number field $K$ equals (naturally?) its TateShafarevich group $\mathrm{III}(A)$.
Is this true? And if so, where can I find a proof?
I read that the BrauerManin obstruction $A(\mathbb{A}_K)^{\mathbf{Br}}$ of an Abelian variety $A$ over a number field $K$ equals (naturally?) its TateShafarevich group $\mathrm{III}(A)$. Is this true? And if so, where can I find a proof? 


The quotient of what you called the BrauerManin 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 BrauerManin obstruction is the closure of $A(K)$. See L. Wang, BrauerManin 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 BrauerManin obstruction usually isn't. 


Manin's ICM talk at Nice (1970) might be a good place to start. If you read French, an elementary introduction to the localtoglobal 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. 

