Section conjecture

In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of the group homomorphism $$\pi_1(X)\to \operatorname{Gal}(k)$$, where $$X$$ is a complete smooth curve of genus at least 2 over a field $$k$$ that is finitely generated over $$\mathbb{Q}$$, in terms of decomposition groups of rational points of $$X$$. The conjecture was introduced by in a 1983 letter to Gerd Faltings.