The section conjecture for rational points on curves over number fields

Let $k$ be a number field, let $X$ be a smooth projective curve over $k$ of genus at least $2$, and fix a geometric point $\bar x$ of $X$. Consider the natural exact sequence

$$1\to \pi_1(\overline X,\bar x)\to \pi_1(X,\bar x)\to \operatorname{Gal}(\overline k|k)\to 1.$$

A continuous section means a continuous homomorphism from $\operatorname{Gal}(\overline k|k)$ to $\pi_1(X,\bar x)$ that is a right inverse to the displayed projection. The section conjecture. The curve $X$ has a $k$-rational point if and only if the map

$$\pi_1(X,\bar x)\to \operatorname{Gal}(\overline k|k)$$

has a continuous section. This is the number-field, projective-curve variant of Grothendieck's section conjecture. The injectivity of the associated map from rational points to conjugacy classes of sections is known, whereas the surjectivity—and hence the asserted equivalence in this formulation—remains widely open.