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 ${\pi }_1(X)\to \mathrm{Gal}(k)$, where $X$ is a complete smooth curve of genus at least 2 over a field $k$ that is finitely generated over $\mathbb{\mathbb{Q}}$, 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.

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.