ProofAtlas

Arithmetic geometry and algebraic differential equations

Grothendieck–Katz p-Curvature Conjecture

Collaboration beta

Does vanishing p-curvature for almost every prime force an algebraic connection to have finite monodromy? The general implication remains open.

ψv=0for almost everyv⇒(E,∇)has finite monodromy.
Known results and sources
A vector bundle connection meets finite-prime flatness motifs beside a characteristic-zero monodromy loop with a visible gap.
The conjecture asks whether almost-everywhere vanishing p-curvature forces finite monodromy; the implication remains open.

Research problem

Exact mathematical statement

Let KK be a number field, X/KX/K a smooth connected algebraic variety, and (E,∇)(E,\nabla) a vector bundle with integrable algebraic connection. The conjecture is

ψv=0for almost every good finite placev⇒(E,∇)has finite monodromy.\psi_v=0\text{ for almost every good finite place }v \quad\Longrightarrow\quad (E,\nabla)\text{ has finite monodromy}.

Problem infographic

Problem at a glance

A connection on an algebraic base is surrounded by many flat prime reductions, separated from an interrupted finite-monodromy target.
The defining connection, its reductions, and the unresolved finite-monodromy question are shown without a rank-specific proof route.

Current mathematical picture

Where work on Grothendieck–Katz p-Curvature Conjecture stands

Partially resolved

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

Useful failureNaive tail-recurrence connection calculation

The negative-index coefficient equation fixes the initial slope and the second exponent-zero branch is logarithmic. A corrected connection calculation using the canonical holomorphic branch and logarithmic companion remains viable.

Route status · Narrowed route
Main reductionCurrent reduction

Prove that the canonical normalized polynomial solution of the fixed rank-six operator has nonzero value at z=1 for every prime p≡13 mod 24.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeCompute the canonical exponent-zero connection coefficient at z=1 with both logarithmic structures controlled.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

Grothendieck–Katz p-Curvature Conjecture in numbers

2.9kretained lines of mathematical investigation2,891 in the current working snapshot
Argument development
2,522 · 87%
Explored or eliminated routes
53 · 2%
Computational analysis
64 · 2%
Open obligations
80 · 3%
Definitions and setup
172 · 6%
7selected mapped statements1routes investigated3open questions3contribution-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 Grothendieck (unpublished origin) and Nicholas M. Katz (precise formulation)Proposed1972Status checkedAug 14, 2026
Source recordedAug 14, 2026

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

11 selected steps

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

11 selected steps

Scroll horizontally to explore the route

Working route overview for Grothendieck–Katz p-Curvature ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Zero p-curvature at almost every finite place should force finite monodromy. — Depends on missing premiseZero p-curvature at almostevery finite place shouldforce…Central finite equivalence — Depends on missing premiseCentral finite equivalenceCurrent reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetExact conjectural implication — Depends on missing premiseExact conjecturalimplicationFinite scans — Depends on missing premiseFinite scansUnique canonical branch — Depends on missing premiseUnique canonical branchNaive tail-recurrence connection calculation — stoppedNaive tail-recurrenceconnection calculationCompute the canonical exponent-zero connection coefficient at z=1 with both logarithmic structures controlled. — OpenCompute the canonicalexponent-zero connectioncoefficient…Show the characteristic-p truncation of that coefficient is a unit for every p congruent to 13 modulo 24. — OpenShow the characteristic-ptruncation of thatcoefficient…Audit the global reduction and complete every boundary and cross-prime step after the core lemma. — OpenAudit the global reductionand complete every boundaryand…
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 routeNaive tail-recurrence connection calculation

The negative-index coefficient equation fixes the initial slope and the second exponent-zero branch is logarithmic. A corrected connection calculation using the canonical holomorphic branch and logarithmic companion remains viable.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Compute the canonical exponent-zero connection coefficient at z=1 with both logarithmic structures controlled.Suggested move: Construct compatible Frobenius bases at zero and one and derive an integral, Barnes, or middle-convolution representation.
Ready to work on
02
Show the characteristic-p truncation of that coefficient is a unit for every p congruent to 13 modulo 24.Suggested move: Reduce the coefficient to an explicit p-adic gamma-unit product or a nonzero Jacobi-sum norm.
Ready to work on
03
Audit the global reduction and complete every boundary and cross-prime step after the core lemma.Suggested move: Treat these five full-conjecture obligations as separate later stages, preserving all imported-theorem hypotheses.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusPartially resolved

The universal p-curvature implication remains open. Important geometric and local-system classes are known, including recent privileged-local-system cases, but they do not establish finite monodromy for every integrable algebraic connection satisfying almost-everywhere zero p-curvature.

[1][2][3]
External progress

What the literature has established

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

  1. PreprintEsnault and Groechenig established the conjecture for a class of privileged local systems, adding higher-dimensional examples without resolving the general statement.[3]
  2. Peer reviewedTang proved a stronger-hypothesis variant over the thrice-punctured projective line and related punctured curves.[2]
  3. Historical sourceKatz gave the foundational precise treatment of p-curvature and algebraic solutions of differential equations.[1]
3 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusGrothendieck–Katz p-Curvature Conjecture
Solved special casestronger-hypothesis punctured-curve variant

A convergence condition stronger than vanishing p-curvature yields trivial monodromy on selected punctured curves.

[2]
Solved special caseprivileged local systems

Privileged local systems form a scoped class for which the conjecture is proved.

[3]

Formalization opportunities

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

  • Formalization targetA formal statement needs models at almost all finite places, p-curvature, algebraic integrable connections, and finite monodromy with exact equivalence to algebraic horizontal sections.
  • Formalization targetAny formal proof must separately import or formalize reduction, rigidity, and arithmetic-geometric inputs; no checked proof of the universal implication was located.

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. 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
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

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.

5 standing statements2 proposed statements3 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction1 of 71
  • lemma3 of 73
  • equivalence1 of 71
  • computational claim1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementZero p-curvature at almost every finite place should force finite monodromy.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact conjectural implicationintermediate
  • retained route statementCentral finite equivalenceintermediate
  • retained route statementUnique canonical branchintermediate
  • retained route statementFinite scansintermediate
  • 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
  • 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 failureNaive tail-recurrence connection calculationreported failure
  • Research targetCompute the canonical exponent-zero connection coefficient at z=1 with both logarithmic structures controlled.open
  • Research targetShow the characteristic-p truncation of that coefficient is a unit for every p congruent to 13 modulo 24.open
  • Research targetAudit the global reduction and complete every boundary and cross-prime step after the core lemma.open
  • Research targetConnection coefficientsuperseded
  • Research targetGlobal completionsuperseded
  • Narrowed routeNaive tail-recurrence connection calculationThe negative-index coefficient equation fixes the initial slope and the second exponent-zero branch is logarithmic. A corrected connection calculation using the canonical holomorphic branch and logarithmic companion remains viable.
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 bridgeCompute the canonical exponent-zero connection coefficient at z=1 with both logarithmic structures controlled.

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 pointCompute the canonical exponent-zero connection coefficient at z=1 with both logarithmic structures controlled.

Grothendieck–Katz p-Curvature Conjecture · ready to start

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Research contextPrepared context for any AI agent

Does vanishing p-curvature for almost every prime force an algebraic connection to have finite monodromy? The general implication remains open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 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. 1
    Algebraic solutions of differential equations (p-curvature and the Hodge filtration)original source · Nicholas M. Katz · Inventiones Mathematicae 18, 1–118 · 1972 · DOI 10.1007/BF01389714 · accessed Aug 14, 2026
  2. 2
    Algebraic solutions of differential equations over P1 − {0,1,∞}peer reviewed result · Yunqing Tang · International Journal of Number Theory 14(5), 1427–1457 · 2018 · DOI 10.1142/S1793042118500884 · accessed Aug 14, 2026
  3. 3
    The Grothendieck-Katz Conjecture for privileged local systemspreprint · Hélène Esnault, Michael Groechenig · arXiv · 2026 · ARXIV 2606.16873 · accessed Aug 14, 2026

Important qualifications

  • External metadata was collected independently of the unreviewed source material; packet claims are not external evidence.
  • Scoped searches do not establish that no additional result, formalization, or dataset exists elsewhere.
  • No external computation was rerun, and no source-submitted URL or packet attachment was fetched, executed, or rendered.
  • This metadata grants no proof, acceptance, credit, publication, redistribution, deployment, or public-write authority.

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