Section conjecture

In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of the group homomorphism ${\displaystyle \pi _{1}(X)\to \operatorname {Gal} (k)}$, where ${\displaystyle X}$ is a complete smooth curve of genus at least 2 over a field ${\displaystyle k}$ that is finitely generated over ${\displaystyle \mathbb {Q} }$, in terms of decomposition groups of rational points of ${\displaystyle X}$. The conjecture was introduced by Alexander Grothendieck (1997) in a 1983 letter to Gerd Faltings.