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Let $X/\mathbb{Z}_p$ be a smooth hyperbolic curve and $\pi^{un}_1(X_{\overline{\mathbb{Q}}_p},b)$ denotes the pro-unipotent completion (over $\mathbb{Q}_p$) of the etale fundamental group of $X$ base changed to $\overline{\mathbb{Q}}_p$, based at a point $b\in X(\mathbb{Z}_p)$.

Then we have a map $$X(\mathbb{Z}_p) \longrightarrow H^1(Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p), \,\pi^{un}_1(X_{\overline{\mathbb{Q}}_p},b)),$$ defined by sending $x\in X(\mathbb{Z}_p)$ to the (cohomology class of) the path torsor $P(b,x)$.

Question. Is this map always injective?

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  • $\begingroup$ Let $J$ be the Jacobian of $X$. Then it's clear that the map from $X$ to $J$ is injective, and the map from $J$ to $H^1(G_{\mathbb{Q}_p},\pi_1^{un}(J_{\overline{\mathbb{Q}_p}},b))$ is injective modulo torsion. But it does seem more tricky to prove that the map on $X(\mathbb{Z}_p)$ is injective. $\endgroup$
    – David Corwin
    Commented Mar 3, 2021 at 1:49
  • $\begingroup$ It's actually not too hard to cook up an example where the map $X(\mathbb{Z}_p) \to H^1(G_{\mathbb{Q}_p},\pi_1^{un}(J_{\overline{\mathbb{Q}_p}},b)) = H^1(G_{\mathbb{Q}_p},\pi_1^{un}(X_{\overline{\mathbb{Q}_p}},b)^{ab})$ is not injective. $\endgroup$
    – David Corwin
    Commented Mar 3, 2021 at 1:50
  • $\begingroup$ The corresponding result for the pro-p fundamental group is at least true, and is a theorem of Mochizuki (Theorem C of The Local Pro-p Anabelian Geometry of Curves). $\endgroup$
    – Alexander Betts
    Commented Apr 5, 2022 at 22:58

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