Let $X_1,X_2$ be two smooth proper schemes over $\mathbb{Z}_p$, such that their reduction modulo $p$ are isomorphic as $\mathbb{F}_p$schemes. Then is it true that the geometry etale fundamental groups $\pi_1^{alg}(X_1\otimes \bar{\mathbb{Q}}_p)$ and $\pi_1^{alg}(X_2\otimes \bar{\mathbb{Q}}_p)$ are isomorphic?
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$\begingroup$ Do you really want to tensor with $\overline{\mathbb{Q}_p}$? In some sense, this "kills the interesting part." I don't know how to turn this into a proof, but it feels like once you're over an algebraically closed field of characteristic $0$ there should be an argument that says your question is equivalent to asking whether or not deformation equivalent varieties over $\mathbb{C}$ have the same topological fundamental group (which sounds true to me). $\endgroup$– MattApr 28, 2014 at 18:28

$\begingroup$ I do want tensor with $\bar{\mathbb{Q}}_p$. Its complex analogue is the following statement: varying complex structure( or even $C^{\infty}$ structure) does not affect topological fundamental group. This is trivially true. But I do not konw wether it is true in padic case. $\endgroup$– LanApr 28, 2014 at 19:43

$\begingroup$ Yes. This is exactly my point. The argument I'm thinking of would go something like, "by the Lefschetz principle ..." but it will take someone other than me to understand whether or not those words can be said in this case. $\endgroup$– MattApr 28, 2014 at 20:34

5$\begingroup$ By Grothendieck's specialisation theorem (in SGA1) the prime to $p$ quotients of both groups are the same, but it seems unlikely to me that both are always isomorphic. (I would guess that there are examples where one group is trivial and the other is $\mathbb{Z}/p$.) $\endgroup$– nafApr 29, 2014 at 3:55

$\begingroup$ (1) For curves, this is okay. (2) If $X_1$ and $X_2$ are hypersurfaces (or more generally complete intersections) in projective space then one could compare degrees to show that generically one is a deformation of the other. $\endgroup$– Manish KumarMay 3, 2014 at 13:24
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