Math Conjectures

Grothendieck Section Conjecture

OPENLandmarkConjectureProposed 1983 · Standard version

Canonical statement

Let KK be a field finitely generated over Q\mathbb Q, put GK=Gal⁡(Kˉ/K)G_K=\operatorname{Gal}(\bar K/K), and let X/KX/K be a smooth, projective, geometrically connected curve of genus g≥2g\ge2. After choosing a geometric base point, its étale fundamental groups fit into the exact sequence
1→π1eˊt(XKˉ)→π1eˊt(X)→GK→1. 1\to\pi_1^{\mathrm{\acute et}}(X_{\bar K}) \to\pi_1^{\mathrm{\acute et}}(X) \to G_K\to1.
The map sending x∈X(K)x\in X(K) to the conjugacy class (under π1eˊt(XKˉ)\pi_1^{\mathrm{\acute et}}(X_{\bar K})) of the section GK→π1eˊt(X)G_K\to\pi_1^{\mathrm{\acute et}}(X) determined by xx is a bijection.
View source LaTeX
Let \(K\) be a field finitely generated over \(\mathbb Q\), put \(G_K=\operatorname{Gal}(\bar K/K)\), and let \(X/K\) be a smooth, projective, geometrically connected curve of genus \(g\ge2\). After choosing a geometric base point, its étale fundamental groups fit into the exact sequence
\[
  1\to\pi_1^{\mathrm{\acute et}}(X_{\bar K})
    \to\pi_1^{\mathrm{\acute et}}(X)
    \to G_K\to1.
\] The map sending \(x\in X(K)\) to the conjugacy class (under \(\pi_1^{\mathrm{\acute et}}(X_{\bar K})\)) of the section \(G_K\to\pi_1^{\mathrm{\acute et}}(X)\) determined by \(x\) is a bijection.
AG-ANAB-005:[Grothendieck1983Faltings][Mochizuki1999Anabelian][Stix2013RationalPoints]
Algebraic geometryNumber theory

Notes

In his 1983 letter to Faltings, Grothendieck proposed that rational points on curves should be governed by the étale fundamental group [Grothendieck1983Faltings]. For a smooth projective geometrically connected curve XX of genus g≥2g\ge2 over a field KK finitely generated over Q\mathbb Q, the group π1eˊt(X)\pi_1^{\mathrm{\acute et}}(X) is an extension of GK=Gal⁡(Kˉ/K)G_K=\operatorname{Gal}(\bar K/K) by the geometric fundamental group, and every point of X(K)X(K) determines a section of this extension, well defined up to conjugacy. The section conjecture asserts that this map from rational points to conjugacy classes of sections is a bijection.

Injectivity has been known for a long time, and there are substantial results in local and birational settings: Mochizuki's local pro-pp anabelian geometry established strong theorems for curves over pp-adic fields [Mochizuki1999Anabelian], and Stix's monograph collects much of what is known around the conjecture [Stix2013RationalPoints].

The essential open part is surjectivity: showing that every section of the extension is point-theoretic, i.e. arises from a rational point. Over number fields this remains unproved, and no general strategy has been carried through.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.