section conjecture

An instance of the anabelian philosophy, Grothendieck’s section conjecture: for a ‘nice’ curve \(X\) over a number field \(F\), the rational points are in bijection with the sections of the exact sequence \begin{align*} 1 \rightarrow \pi_1^\text{ét}(X_{\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}) \rightarrow \pi_1^ \text{ét}(X) \rightarrow { \mathsf{Gal}} (F^s/F) \rightarrow 1 \end{align*}

Still wide open, a proof would allow checking arithmetically interesting things like existence of rational points on curves by analyzing maps between homotopy groups.