The local p-adic section conjecture for curves of genus at least two

Claims Theorem 1.1: for every finite extension $k/\mathbb Q_p$ and every smooth proper geometrically connected curve $X/k$ of genus at least two, $X(k)\to\{\text{sections of }\pi_1(X)\to G_k\}/\pi_1(X_{\bar k})$ is bijective. Through Theorem 1.2 every section localized at a rank-one valuation lifts to a birational section. It does NOT prove the global section conjecture over number fields in general (see the separate entry for the cases it does reach), and says nothing about affine hyperbolic curves or curves over other local fields.

Claims Theorem 1.1: for every finite extension $k/\mathbb Q_p$ and every smooth proper geometrically connected curve $X/k$ of genus at least two, $X(k)\to\{\text{sections of }\pi_1(X)\to G_k\}/\pi_1(X_{\bar k})$ is bijective. Through Theorem 1.2 every section localized at a rank-one valuation lifts to a birational section. It does NOT prove the global section conjecture over number fields in general (see the separate entry for the cases it does reach), and says nothing about affine hyperbolic curves or curves over other local fields.