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I would like to know whether, given an elliptic curve $E$ over $\mathbb{Q}$, there is a "natural" topological space associated to $E$ whose fundamental group is (isomorphic to) the Tate-Shafarevich group of $E$ or not. Thank you in advance.

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Tate-Shafarevich groups are always abelian. Isn't the abelianisation of a fundamental group usually a first homology group? And with trivial coefficients as well. But now that looks a bit strange because the Tate-Shafarevich group is a (subspace of a) 1st *co*homology group, and with non-trivial coefficients too. So I am not at all optimistic. –  Kevin Buzzard Apr 28 '11 at 19:46
Yes, but there is a theorem that says that every group $G$ is the fundamental group of some topological space. The question is: among all the topological spaces whose fundamental group is (isomorphic to) $Sha(E)$, is there one which reflects quite faithfully the properties of $E$? –  Sylvain JULIEN Apr 28 '11 at 20:31
One could ask that question about any group which appears in mathematics. Is there a particular reason to expect in this case that there is a reasonable answer? –  Qiaochu Yuan Apr 29 '11 at 6:03
Perhaps, but "any group which appears in mathematics" may not be as relevant when one deals with rational elliptic curves as this one. Moreover, the fact that most people think that Sha(E/Q) is finite (with order the square on an integer) might be important in such a framework, since subgroups of the fundamental group of a topological space are in correspondance with its covering spaces. –  Sylvain JULIEN May 12 '11 at 23:04
I obviously meant "correspondence" in the previous comment, I apologize for such an ugly gallicism. –  Sylvain JULIEN May 12 '11 at 23:07
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