Rational points · Fano varieties · height asymptotics · thin exceptional sets

Manin–Peyre Conjecture on Rational Points

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After removing a geometrically defined thin exceptional set, do rational points of bounded anticanonical height on a suitable Fano variety follow the predicted Peyre asymptotic?

N_{X, H} (B) \sim c_{X, H} B (\log B)^{ρ (X) - 1}

Known results and sources

Manin–Peyre predicts the leading distribution of rational points after the correct thin exceptional set is removed.

Research problem

Exact mathematical statement

In the modern thin-set Manin–Peyre formulation, let $X$ be a suitable Fano variety over a number field $F$ , let $H_{-K_X}$ be an anticanonical height, and remove a prescribed thin subset $T\subset X(F)$ . The conjectural counting function

N_{X,H_{-K_X}}(B)=\#\{x\in X(F)\setminus T:H_{-K_X}(x)\le B\}

has the Peyre-predicted leading asymptotic, customarily of the form

N_{X,H_{-K_X}}(B)\sim c_{X,H} B(\log B)^{\rho(X)-1},

under the formulation's stated geometric and arithmetic hypotheses. A thin subset of $X(F)$ is contained in a finite union of type-I contributions from rational points on proper closed subvarieties and type-II images $f(Y(F))$ of dominant generically finite morphisms $f:Y\to X$ of degree at least two. The anticanonical exponent displayed here must not be transferred unchanged to an arbitrary ample height, whose general formulation uses its own $a(X,L)$ and $b(F,X,L)$ invariants. The source packet does not claim the general statement. Its concrete stress test is

x_0y_0^3+x_1y_1^3+x_2y_2^3+x_3y_3^3=0\subset\mathbf P_x^3\times\mathbf P_y^3,

and the complete height-cone asymptotic for that fivefold also remains unproved.

Problem infographic

Problem at a glance

A problem-first diagram shows a Fano variety over F, anticanonical-height rational points outside a thin subset T of X(F), the Peyre growth law, and a separate arbitrary-height stress-test formulation using a(X,L) and b(F,X,L). — For anticanonical height, Manin–Peyre predicts the rational-point asymptotic outside a thin subset of X(F); arbitrary ample-height formulations use different a- and b-invariants.

Current mathematical picture

Where work on Manin–Peyre Conjecture on Rational Points stands

Open conjecture

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

Useful failureAbsolute rearrangement of the Möbius sum without sign cancellation

The source computes a relative loss R^(3/2)/P and calls this a route-elimination certificate. Signed Möbius-kernel cancellation, central/counterterm recombination, primal relation-lattice variables, summing over y before planes, and a secondary-term interpretation remain proposed.

Route status · Narrowed route

Main reductionCurrent reduction

For the cubic-bundle fivefold stress test, the current work groups the full-support dual remainder by primitive planes and reduces the unproved portion to a signed combination of central, counterterm, and transition contributions.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve cancellation or structural recombination for the combined central and counterterm packet.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Manin–Peyre Conjecture on Rational Points in numbers

4.2kretained lines of mathematical investigation4,163 in the current working snapshot

Argument development: 3,622 · 87%
Explored or eliminated routes: 78 · 2%
Computational analysis: 25 · 1%
Open obligations: 211 · 5%
Definitions and setup: 227 · 5%

8selected mapped statements1routes investigated4open questions4contribution-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 byJens Franke, Yuri I. Manin, Yuri TschinkelProposed1989Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Prove cancellation or structural recombination for the combined central and counterterm packet.

Suggested move: Retain the exact cutoff and test the signed kernels only after the finite d-sum, with every normalization and cone restriction explicit.

What would count as progress

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

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

13 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.

Explored alternatives

Other routes

1 recorded

Narrowed routeAbsolute rearrangement of the Möbius sum without sign cancellation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Prove cancellation or structural recombination for the combined central and counterterm packet.Suggested move: Retain the exact cutoff and test the signed kernels only after the finite d-sum, with every normalization and cone restriction explicit.

Ready to work on

Control the localized transition discrepancy on its effective determinant band.Suggested move: Seek a summable saving after the d-sum while retaining the dependence d approximately P Delta_N divided by D_y.

Ready to work on

Combined signed estimate

The live open bridge is a uniform little-o estimate for the combined central, counterterm, and transition contribution through the middle and small-P cones.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Globalize any dyadic gain without upgrading the sector theorem to the complete fivefold or general conjecture.Suggested move: Audit the inherited globalization hypotheses and integrate the middle and small-P cones with a uniform summable power saving.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

The modern geometrically corrected thin-set Manin–Peyre conjecture remains open in general. The literature proves many families and constrains exceptional contributions, but neither those cases nor the featured cubic-bundle fivefold settle the universal statement.

[1][2]

External progress

What the literature has established

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

2022
Peer reviewedLehmann, Sengupta, and Tanimoto proposed a geometric thin exceptional set and proved the relevant larger-invariant contributions lie in a thin subset.[2]
1989
Historical sourceFranke, Manin, and Tschinkel initiated the bounded-height asymptotic framework for rational points on Fano varieties.[1]

Mathematical neighborhood

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. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Corrected the research recordCorrection note

Correction details

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.

6 standing statements2 proposed statements4 open questions1 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction3 of 83
lemma2 of 82
equivalence1 of 81
negative result1 of 81

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included

retained route statementThe rational-point count has the predicted geometric and adelic leading term.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementConcrete fivefoldintermediate
retained route statementAudit-pending incidence boundintermediate
retained route statementTransferred sector resultintermediate
retained route statementExact full-support organizationintermediate
retained route statementAbsolute-envelope barrierintermediate
Recorded relationshipThe source 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 source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureAbsolute rearrangement of the Möbius sum without sign cancellationreported failure
Research targetProve cancellation or structural recombination for the combined central and counterterm packet.open
Research targetControl the localized transition discrepancy on its effective determinant band.open
Research targetGlobalize any dyadic gain without upgrading the sector theorem to the complete fivefold or general conjecture.open
Research targetCombined signed estimateopen
Narrowed routeAbsolute rearrangement of the Möbius sum without sign cancellationThe source computes a relative loss R^(3/2)/P and calls this a route-elimination certificate. Signed Möbius-kernel cancellation, central/counterterm recombination, primal relation-lattice variables, summing over y before planes, and a secondary-term interpretation remain proposed.

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 bridgeProve cancellation or structural recombination for the combined central and counterterm packet.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

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.

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Prepared starting pointProve cancellation or structural recombination for the combined central and counterterm packet.

Manin–Peyre Conjecture on Rational Points · ready to start

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Research contextPrepared context for any AI agent

After removing a geometrically defined thin exceptional set, do rational points of bounded anticanonical height on a suitable Fano variety follow the predicted Peyre asymptotic?

Exact question and boundaries
Current routes and known obstacles
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Sources and references2 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
Rational points of bounded height on Fano varietiesoriginal source · Jens Franke, Yuri I. Manin, Yuri Tschinkel · Inventiones Mathematicae · 1989 · DOI 10.1007/BF01393904 · accessed Aug 14, 2026
2
Geometric consistency of Manin's Conjecturepeer reviewed result · Brian Lehmann, Akash Kumar Sengupta, Sho Tanimoto · Compositio Mathematica · 2022 · ARXIV 1805.10580 · DOI 10.1112/S0010437X22007588 · accessed Aug 14, 2026

Important qualifications

External status was checked only against the listed primary, peer-reviewed, official-publisher, or author-hosted sources; omission is not a global nonexistence claim.
Special cases, reductions, preprints, and source-reported claims are kept at their exact scope and do not inherit proof, review, acceptance, publication, or priority authority.
The scoped formalization check found no problem-level statement or proof in current Mathlib documentation or the public Formal Conjectures repository; this is not evidence that no formalization exists elsewhere.
No source material, attachment, submitted URL, or packet computation was executed, rendered, fetched, or used as external authority.

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Manin–Peyre Conjecture on Rational Points

Exact mathematical statement

Problem at a glance

Where work on Manin–Peyre Conjecture on Rational Points stands

Manin–Peyre Conjecture on Rational Points in numbers

Prove cancellation or structural recombination for the combined central and counterterm packet.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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