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This work shows the de Jong fundamental group of non-trivial abelian varieties over complete algebraically closed extensions is non-abelian, suggesting significant implications for algebraic geometry.
We show that the de Jong fundamental group of any non-trivial abelian variety over a complete algebraically closed extension C/Qₚ is non-abelian. Generalizing an argument for P¹C, we also show that the de Jong fundamental group of any connected rigid analytic variety over C admitting a non-constant map to PⁿC (e.g., an abelian variety) depends on C and is big.
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Sean Howe (2025) studied this question.
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