---
date: 2022-04-05 23:42
modification date: Sunday 24th April 2022 16:35:18
title: "section conjecture"
aliases: [section conjecture]
---

Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #open/conjectures #open/problems 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

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# section conjecture


 An instance of the [anabelian](anabelian.md) philosophy, Grothendieck's **section conjecture**: for a 'nice' [curve](Unsorted/curves.md) $X$ over a [number field](Unsorted/number%20field.md) $F$, the [rational points](rational points) are in bijection with the sections of the exact sequence
$$
1 \rightarrow \pi_1^\et(X_{\bar{F}}) \rightarrow \pi_1^
\et(X) \rightarrow \Gal(F^s/F) \rightarrow 1
$$

Still wide open, a proof would allow checking arithmetically interesting things like existence of [rational points](Unsorted/rational%20points.md) on [curves](Unsorted/curves.md) by analyzing maps between homotopy groups.