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section conjecture

Idea

Let $k$ be a field of characteristic zero with algebraic closure $\overline{k}$ and absolute Galois group ${\Gamma }_k:=\mathrm{Gal}(\overline{k}/k)$. Let $X$ be a geometrically connected variety over $k$. Fix a geometric point, $\overline{x}\in X(\overline{k})$ and let ${\pi }_1(X):={\pi }_1(X,\overline{x})$ be the étale fundamental group of $X$. Set $\overline{X}=X{\times }_k\overline{k}$ and denote by ${\pi }_1(\overline{X}):={\pi }_1(\overline{X},\overline{x})$ the étale fundamental group of $\overline{X}$.

Grothendieck’s fundamental exact sequence of profinite groups, cf. SGA1 IX Thm 6.1, (see also SGA1), is:

$$1\to {\pi }_1(\overline{X})\to {\pi }_1(X)\to {\Gamma }_k\to 1.$$1 \to\pi_1 (\overline{X})\to\pi_1 (X) \to \Gamma_k \to 1.

By functoriality of ${\pi }_1$, the existence of a $k$-point on $X$ implies that the above exact sequence has a section.

Grothendieck’s section conjecture

This predicts that the converse statement is true whenever $X$ is a proper hyperbolic curve over a number field.

(Needs more from Galois Theory and Diophantine geometry below)

Blog discussions

• n-Category café

• Motivic stuff: Cohomology, homotopy theory, and arithmetic geometry (Andreas Holmstrom on October 4, 2009)

Related entries

• anabelian geometry

References

• Mohamed Saidi, Around the Grothendieck Anabelian Section Conjecture, ArXiv:1010.1314v2.

• Minhyong Kim, Galois Theory and Diophantine geometry