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Let $X$ be smooth projective variety, such as an elliptic curve. We know that $X$ does not have a universal cover in the category of schemes.

However, if $k = \mathbf{C}$, then $\tilde{X}$ exists as a complex manifold. For example, $\tilde{X} \cong \mathbf{C} \longrightarrow X \cong \mathbf{C}/\Lambda$ for an elliptic curve.

What about $k = \mathbf{F}_q$ or $k = \mathbf{Q}_p$? Does a universal cover for $X$ exist in some appropriate$^\ast$ category?

($\ast$) One which interacts with the algebraic geometry of $X$ in a meaningful way.

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  • $\begingroup$ something like Tate uniformization is potentially relevant. $\endgroup$
    – user145520
    Commented Nov 8, 2019 at 1:27
  • $\begingroup$ Sixty years ago Tate showed the answer is yes for an elliptic curve over $\mathbb{Q}_p$ whose reduction mod p has a node with tangents rational over the base field. Since then there has been a huge amount of work. For a brief friendly introduction, see Section 3 of Milne's article "The Work of John Tate", available on his webpage jmilne.org/math/ under Expository Notes. $\endgroup$
    – anon
    Commented Nov 9, 2019 at 13:40
  • $\begingroup$ So there is no general construction yet for a smooth projective scheme over $\mathbf{Q}_q$? Is there any consensus on whether such a thing should exist? $\endgroup$
    – Kim
    Commented Nov 9, 2019 at 22:35
  • $\begingroup$ @anon's link, clickably: Milne - The work of John Tate. $\endgroup$
    – LSpice
    Commented Nov 10, 2019 at 0:21
  • $\begingroup$ You may enjoy: arxiv.org/abs/0902.3464 $\endgroup$
    – Daniel Litt
    Commented Nov 10, 2019 at 0:21

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