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 Alexander Grothendieck (1997) in a 1983 letter to Gerd Faltings.