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 AlexanderGrothendieck (1997) in a 1983 letter to Gerd Faltings.

• Grothendieck, Alexander (1997), "Brief an G. Faltings", in Schneps, Leila; Lochak, Pierre (eds.), Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser., vol. 242, Cambridge University Press, pp. 49–58, ISBN 978-0-521-59642-8, MR 1483108

• "Why is the section conjecture important?". mathoverflow.net.