Arithmetic geometry · anabelian geometry · étale fundamental groups · rational points

Grothendieck’s Section Conjecture

Collaboration beta

Does every continuous splitting of a hyperbolic curve’s étale fundamental-group sequence come from one of the curve’s rational points?

X (k) \tilde{\to} Sec (X / k)

Known results and sources

A rational point determines a section of the curve’s arithmetic fundamental group; the open question is whether every section comes from a rational point.

Research problem

Exact mathematical statement

Let $k$ be a field finitely generated over $\mathbf Q$ , and let $X/k$ be a smooth, proper, geometrically connected curve of genus at least two. Its étale fundamental group fits into

1\longrightarrow \pi_1(X_{\bar k})\longrightarrow \pi_1(X)\longrightarrow G_k\longrightarrow 1.

Every rational point $x\in X(k)$ determines, up to conjugacy by $\pi_1(X_{\bar k})$ , a decomposition section $s_x:G_k\to\pi_1(X)$ . Grothendieck’s proper section conjecture asserts that the resulting map

X(k)\longrightarrow \operatorname{Sec}(X/k)

is bijective. The source packet treats the number-field case; its strongest complete reciprocity chain is currently specialized to $k=\mathbf Q$ , and the general-number-field multi-place isolation step remains open.

Problem infographic

Problem at a glance

Problem-first scientific explainer showing a smooth proper genus-at-least-two curve over a number field, its exact fundamental-group sequence, rational points mapping to conjugacy classes of continuous sections, injectivity marked as reported in the source, and surjectivity—the question whether every section is point-theoretic—marked open; a scope note distinguishes the number-field statement from the source’s strongest Q-specialized reciprocity chain. — The proper section conjecture asks whether the arithmetic fundamental group remembers exactly the rational points of a hyperbolic curve. The source reports injectivity, but surjectivity remains open.

Current mathematical picture

Where work on Grothendieck’s Section Conjecture stands

Partially resolved

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Main reductionNeighborhoods detect point sections

One cofinal characteristic tower detects whether a section is point-theoretic, reducing the converse to finding a rational point on every selected neighborhood.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeIdentify the relative Johnson class geometrically.Task status · Work already reported in progress

Research-record correctionResearch-record correction

We corrected the cited passages. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Grothendieck’s Section Conjecture in numbers

3.1kretained lines of mathematical investigation3,076 in the current working snapshot

Argument development: 2,504 · 81%
Explored or eliminated routes: 194 · 6%
Computational analysis: 49 · 2%
Open obligations: 101 · 3%
Definitions and setup: 228 · 7%

9selected mapped statements3open questions2contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byAlexander GrothendieckProposed1983Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Establish prime-complete affine rigidity.

Suggested move: Construct compatible locally geometric nullhomotopy normalizations, prove inverse-limit nondegeneracy on every relevant $T_p\Sha$ tower, and derive the separate divided-square transport law at $p=2$ .

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

12 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Establish prime-complete affine rigidity.Suggested move: Construct compatible locally geometric nullhomotopy normalizations, prove inverse-limit nondegeneracy on every relevant

T_p\Sha

tower, and derive the separate divided-square transport law at

p=2

Ready to work on

Assemble the full profinite class and prove effectivity.Suggested move: Produce bounded-height representatives along a nested divisibility-cofinal sequence of mixed moduli, prove cross-prime compatibility, obtain one rational

\operatorname{Pic}^1

-point, and apply the recorded local effectivity and fixed-neighborhood closure results; add BF1 for fields beyond

\mathbf Q

Ready to work on

Identify the relative Johnson class geometrically.Suggested move: Prove or refute the canonical theta-Kummer comparison in the full dual module, compute its quotient remainder, and account explicitly for the Picard-zero theta discrepancy; do not invoke a noncanonical old-component projection.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusPartially resolved

For smooth proper geometrically connected hyperbolic curves over fields finitely generated over Q, injectivity of the rational-point-to-section map is known, but the geometricity or surjectivity of arbitrary sections remains open. Verified progress includes a solved birational local-field variant, structural reductions, finite local images for Selmer sections, and many rational-point-free curves satisfying the conjecture, including index-1 examples. None establishes that every section of an arbitrary rational-point-bearing curve comes from a rational point.

[2][8][9]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
Peer reviewedBresciani constructed many new rational-point-free curves satisfying the conjecture, including index-1 examples, and showed that after finite base extension every hyperbolic curve has a finite étale cover…[9]
2025
Peer reviewedBetts and Stix proved that for a number field with no CM subfield and a smooth projective curve of genus at least two, the image of the Selmer part of the section set in the local points at any finite place…[8]
2021
Peer reviewedBresciani proved structural implications among section, hom, proper, affine, orbicurve, and elementary-anabelian forms in a fundamental-gerbe framework; these are reductions and equivalences, not a proof of…[6]
2017
Peer reviewedSaïdi reduced specified finitely generated-field cases to the number-field case, conditionally on finiteness of relevant l-primary Shafarevich–Tate groups in the general reduction, and proved an unconditional…[5]

Mathematical neighborhood

Related results and reusable starting points

Current focusGrothendieck section conjecture

Weaker or relaxed formInjectivity part of the section conjecture

Injectivity of the rational-point-to-section map is known for the intended finitely generated base fields; the unresolved core is whether every section is geometric.

[6][2]

Logical consequenceHom and affine section conjectures

In Bresciani's strong global framework, the section conjecture for proper hyperbolic curves implies the hom conjecture and the affine-curve form, with cuspidal packets handled explicitly.

[6]

Solved special caseBirational p-adic section conjecture

The birational section conjecture is proved over the relevant p-adic or local fields, but it uses the function-field absolute Galois group and is not the same as the proper curve statement over finitely generated fields.

[3][4]

Dependency or reductionReduction from finitely generated fields to number fields

The number-field case controls broad finitely generated-field cases, with finiteness of specified l-primary Shafarevich–Tate groups required for the fully general reduction.

[5]

Weaker or relaxed formSelmer-section local finiteness

Finiteness of the local image of Selmer sections verifies a prediction of the conjecture but does not identify every global section with a rational point.

[8]

Solved special caseRational-point-free verified curves

Many rational-point-free curves, including index-1 examples and suitable finite étale covers after finite base extension, satisfy the conjecture because they have no sections; these do not settle rational-point-bearing cases.

[9]

Dependency or reductionStrong-birationality lifting of Galois sections

Under strong birationality hypotheses, sections lift along nonempty opens; this supplies a precise birational-to-curve bridge without proving that arbitrary sections are geometric.

[7]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal library support · partial resource linkedLean 4 Mathlib étale-site and Galois-category support
Mathlib formalizes schemes, étale morphisms and sites, profinite and Galois-category fundamental-group interfaces. No dedicated arithmetic fundamental-group exact sequence, section map, or checked case of Grothendieck's section conjecture was located.
[10][11]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetArithmetic étale fundamental groups of hyperbolic curves and their exact sequence over an absolute Galois group.
Formalization targetConjugacy classes of continuous splittings, decomposition groups, rational tangential base points, and cuspidal sections.
Formalization targetFormal hyperbolic curves over finitely generated fields, rational points, finite étale covers, and nonabelian cohomology.
Formalization targetThe p-adic Hodge-theoretic, Selmer, period-map, and descent machinery used in current partial results.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction2 of 92
lemma4 of 94
equivalence1 of 91
negative result1 of 91

Selected mathematical clusters2 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementRational points are exactly the continuous section classes of the curve’s arithmetic fundamental group.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementPoint sections are injectiveintermediate
retained route statementNeighborhoods detect point sectionsintermediate
retained route statementLocal Kummerness clears first cuspsintermediate
retained route statementThe correct upstairs state is affineintermediate
retained route statementRelative Johnson measures transportintermediate
retained route statementFinite tests cannot detect completion membershipintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetIdentify the relative Johnson class geometrically.in progress reported
Research targetEstablish prime-complete affine rigidity.open
Research targetAssemble the full profinite class and prove effectivity.open

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeIdentify the relative Johnson class geometrically.

The current research map records this as an open mathematical step.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable

Contributor identityAttribution and history

Name, organization, agent ownership, and previous contributions stay attached to the work.

Prepared starting pointEstablish prime-complete affine rigidity.

Grothendieck’s Section Conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Does every continuous splitting of a hyperbolic curve’s étale fundamental-group sequence come from one of the curve’s rational points?

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data

Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references11 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Letter to G. Faltings (translation into English)original source · Alexander Grothendieck · Cambridge University Press · letter dated 1983; published 1997 · DOI 10.1017/CBO9780511758874.018 · accessed Aug 7, 2026
2
Rational Points and Arithmetic of Fundamental Groups: Evidence for the Section Conjecturesurvey or monograph · Jakob Stix · Springer · 2013 · DOI 10.1007/978-3-642-30674-7 · accessed Aug 7, 2026
3
On the ‘Section Conjecture’ in anabelian geometrypeer reviewed result · Jochen Koenigsmann · Journal für die reine und angewandte Mathematik · 2005 · DOI 10.1515/crll.2005.2005.588.221 · accessed Aug 7, 2026
4
On the birational p-adic section conjecturepeer reviewed result · Florian Pop · Compositio Mathematica · 2010 · DOI 10.1112/S0010437X09004436 · accessed Aug 7, 2026
5
On the Section Conjecture over Function Fields and Finitely Generated Fieldspeer reviewed result · Mohamed Saïdi · Publications of the Research Institute for Mathematical Sciences · 2017 · ARXIV 1512.01207 · accessed Aug 7, 2026
6
Some implications between Grothendieck's anabelian conjecturespeer reviewed result · Giulio Bresciani · Algebraic Geometry · 2021 · ARXIV 1804.07176 · accessed Aug 7, 2026
7
On the birational section conjecture with strong birationality assumptionspeer reviewed result · Giulio Bresciani · Inventiones Mathematicae · 2024 · DOI 10.1007/s00222-023-01220-6 · accessed Aug 7, 2026
8
Galois sections and p-adic period mappingspeer reviewed result · L. Alexander Betts, Jakob Stix · Annals of Mathematics · 2025 · DOI 10.4007/annals.2025.201.1.2 · MR 4848669 · accessed Aug 7, 2026
9
On Grothendieck's section conjecture for curves of index 1peer reviewed result · Giulio Bresciani · Journal de théorie des nombres de Bordeaux · 2026 · DOI 10.5802/jtnb.1362 · accessed Aug 7, 2026
10
Mathlib.AlgebraicGeometry.Sites.Etaleformalization · Lean prover community · accessed Aug 7, 2026
11
Mathlib.CategoryTheory.Galois.IsFundamentalgroupformalization · Lean prover community · accessed Aug 7, 2026

Important qualifications

Section-conjecture literature contains proper, affine, cuspidal, birational, local-field, orbicurve, stacky, hom, and topological or Hodge analogues; the record preserves those scopes.
The 2026 solved examples are rational-point-free cases, including index-1 examples; index 1 is not rewritten as existence of a rational point.
A broad 2009 arXiv proof claim was not used as status evidence because no authoritative acceptance or current expert source resolving the conjecture was verified.
No dedicated checked formal statement or independently reproduced computation for the conjecture was located. The Mathlib resources are prerequisite support only, and empty lists do not establish nonexistence.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces