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arXiv · 2604.24898

Elementary anabelian varieties are anabelian

Magnus Carlson
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Abstract

We show that isomorphisms of fundamental groups of elementary anabelian varieties -- varieties obtained as iterated fibrations of hyperbolic curves -- over sub-$p$-adic fields correspond bijectively to isomorphisms of varieties. Moreover, dominant maps between proper elementary anabelian varieties are in bijection with ``stably cohomologically injective'' maps of fundamental groups: open maps whose pullbacks to all open subgroups induce injections on cohomology rings with $\ell$-adic coefficients, for any prime $\ell$. This verifies conjectures of Grothendieck from his letter to Faltings. Finally, we establish \'etale homotopical generalizations of these results.

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Magnus Carlson. 2026-04-27. Elementary anabelian varieties are anabelian. https://arxiv.org/abs/2604.24898

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