I don't think there is any reason to think that one exists, especially because the analogy is not very tight. For example, if $X_K = \operatorname{Spec}(\mathcal{O}_K)$, there exist closed hyperbolic 3-manifolds $M$ such that the abelianization of the fundamental group is infinite. (In fact, one conjectures that all hyperbolic $M$ virtually (= after passing to a finite cover) have this property.) On the other hand, the abelianization of $\pi_1(X_K)$ is always finite, by class field theory. As has been remarked elsewhere, there are several non-trivial $K$ such that $pi_1(X_K)$ is trivial.

I don't think there is any reason to think that one exists, especially because the analogy is not very tight. For example, if $X_K = \operatorname{ Spec }(\mathcal{ O }_K)$, there exist closed hyperbolic 3-manifolds $M$ such that the abelianization of the fundamental group is infinite. (In fact, one conjectures that all hyperbolic $M$ virtually (= after passing to a finite cover) have this property.) On the other hand, the abelianization of $\pi_1(X_K)$ is always finite, by class field theory. As has been remarked elsewhere, there are several non-trivial $K$ such that $\pi_1( X_K )$ is trivial.

I don't think there is any reason to think that one exists, especially because the analogy is not very tight. For example, if X_K = Spec(O_K), there exist closed hyperbolic 3-manifolds M such that the abelianization of the fundamental group is infinite. (In fact, one conjectures that all hyperbolic M virtually (= after passing to a finite cover) have this property.) On the other hand, the abelianization of pi_1(X_K) is always finite, by class field theory. As has been remarked elsewhere, there are several non-trivial K such that pi_1(X_K) is trivial.

I don't think there is any reason to think that one exists, especially because the analogy is not very tight. For example, if $X_K = \operatorname{Spec}(\mathcal{O}_K)$, there exist closed hyperbolic 3-manifolds $M$ such that the abelianization of the fundamental group is infinite. (In fact, one conjectures that all hyperbolic $M$ virtually (= after passing to a finite cover) have this property.) On the other hand, the abelianization of $\pi_1(X_K)$ is always finite, by class field theory. As has been remarked elsewhere, there are several non-trivial $K$ such that $pi_1(X_K)$ is trivial.