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Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.

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Questions about analogy between Spec Z and 3-manifolds

The analogy doesn't quite give a number theoretic version of the Poincare conjecture. See Sikora, "Analogies between group actions on 3-manifolds and number fields" (arXiv:0107210): the author states …
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