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This work shows that the de Jong fundamental group depends on the field, suggesting complex behaviors in algebraically closed fields.
For C/Qₚ complete and algebraically closed, we show that the de Jong fundamental group π1,dJ(P¹C) depends on C and, if C has cardinality >2N, that it is not topologically countably generated. The result and proofs generalize to any connected rigid analytic variety with a non-constant map to PⁿC.
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Sean Howe (2025) studied this question.
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