Section conjecture

Short description: A conjecture of Grothendieck

In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of the group homomorphism [math]\displaystyle{ \pi_1(X)\to \operatorname{Gal}(k) }[/math], where [math]\displaystyle{ X }[/math] is a complete smooth curve of genus at least 2 over a field [math]\displaystyle{ k }[/math] that is finitely generated over [math]\displaystyle{ \mathbb{Q} }[/math], in terms of decomposition groups of rational points of [math]\displaystyle{ X }[/math]. The conjecture was introduced by Alexander Grothendieck (1997) in a 1983 letter to Gerd Faltings.

References

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

External links

• "Why is the section conjecture important?". https://mathoverflow.net/q/179664.