Wikipedia geometry item 17: Section conjecture on splittings of group homomorphisms from fundamental groups of complete smo…

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $k$ be a field finitely generated over $\mathbb Q$, fix an algebraic closure $\bar k$, and let $X/k$ be a smooth, proper, geometrically connected curve of genus at least two. Write $X_{\bar k}=X\times_k\bar k$, and choose a geometric base point $\bar x$. The étale fundamental groups fit into the exact sequence of profinite groups \[1\longrightarrow\pi_1(X_{\bar k},\bar x)\longrightarrow\pi_1(X,\bar x)\mathrel{\mathop{\longrightarrow}^{p}}\operatorname{Gal}(\bar k/k)\longrightarrow1.\] A section is a continuous homomorphism $s$ with $p\circ s=\mathrm{id}$. Two sections are equivalent if conjugate by an element of $\pi_1(X_{\bar k},\bar x)$. Every rational point $x\in X(k)$ induces a section class $[s_x]$, independently of the path used to compare base points.

Statement recorded in the supplied source.

Is the map $X(k)\longrightarrow\{\text{continuous sections of }p\}/\pi_1(X_{\bar k},\bar x)$, $x\longmapsto[s_x]$, a bijection for every such $k$ and $X$? [S2]

Short English statement

For a complete hyperbolic curve over a field finitely generated over the rationals, do its rational points correspond exactly to the splittings of its arithmetic fundamental-group sequence, up to conjugacy by the geometric fundamental group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200017 (AMR-071-0017). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hélène Esnault and Phùng Hô Hai, Packets in Grothendieck’s Section Conjecture (2007). Section 2, exact sequence (2.2) and the complete-curve formulation (SC), pp. 3–4. https://arxiv.org/pdf/math/0703877v2.