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