G-functions · arithmetic differential equations · Picard–Fuchs operators · motives

Bombieri–Dwork Conjecture for G-functions

Collaboration beta

Does every minimal differential equation of a G-function arise from algebraic geometry through Gauss–Manin or Picard–Fuchs constructions?

{G-operators} = {operators of geometric origin}

Known results and sources

Bombieri–Dwork asks whether every G-operator belongs to the differential world generated by geometry.

Research problem

Exact mathematical statement

A power series

f(z)=\sum_{n\ge0}a_nz^n\in\overline{\mathbf Q}[[z]]

is a G-function when it is holonomic, all coefficients lie in one number field with every conjugate $|\sigma(a_n)|$ bounded exponentially, and there are positive integers $d_n$ , themselves bounded exponentially, such that $d_n a_m$ is an algebraic integer for every $0\le m\le n$ . Its minimal annihilator is a G-operator. The Bombieri–Dwork conjecture asks whether every G-operator is of geometric origin: equivalently, whether its differential module belongs to the differential category generated, under the standard geometric operations, by subquotients and extensions of Gauss–Manin modules, or by factors of Picard–Fuchs operators. The source explicitly does not prove this statement.

Problem infographic

Problem at a glance

A problem-first diagram states that all coefficients lie in one number field, gives the three G-function conditions including one denominator clearing every coefficient a_0 through a_n, then distinguishes the known geometry-to-G-operator direction from the open converse. — Geometry produces G-operators in the known direction; Bombieri–Dwork asks for the converse, with the full common-denominator condition retained.

Current mathematical picture

Where work on Bombieri–Dwork Conjecture for G-functions stands

Open conjecture

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

Useful failurePromoting first-column or local Cartier success to a full Frobenius extension

The current work gives separate second-stage, Hodge-source, and global H1 adversarial examples. A calibrated geometric example followed by canonical Galochkin–Cartier comparison, all-order uniform lifting, higher-rank reconstruction, and geometric recognition remains the stated route.

Route status · Narrowed route

Main reductionG-operator target

The source defines a G-operator as the minimal differential operator annihilating a G-function.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProduce a fully normalized geometric calibration example with an independently known Frobenius matrix.Task status · Ready to work on

Research-record correctionResearch-record correction

We 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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Bombieri–Dwork Conjecture for G-functions in numbers

2.4kretained lines of mathematical investigation2,380 in the current working snapshot

Argument development: 2,013 · 85%
Explored or eliminated routes: 82 · 3%
Computational analysis: 42 · 2%
Open obligations: 49 · 2%
Definitions and setup: 194 · 8%

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 byEnrico Bombieri, Bernard DworkStatus checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Produce a fully normalized geometric calibration example with an independently known Frobenius matrix.

Suggested move: Fix the Frobenius lift, residue untwisting, graded lines, and divided normalization before comparing source and Cartier coordinates.

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 routePromoting first-column or local Cartier success to a full Frobenius extension

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Produce a fully normalized geometric calibration example with an independently known Frobenius matrix.Suggested move: Fix the Frobenius lift, residue untwisting, graded lines, and divided normalization before comparing source and Cartier coordinates.

Ready to work on

Construct an invertible rank-two semilinear Frobenius matrix with uniform dagger radius and matching local and global extension data.Suggested move: Solve every displayed column equation, Witt–Cartier divisibility, and global Cech class in the same normalized lattice.

Ready to work on

Normalized weak-link bridge

The proposed identification of the weak-link scalar with the local Hodge–Cartier mismatch remains unproved and normalization-dependent.

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

Ready to work on

Extend to higher rank and prove a geometric-recognition theorem compatible with conjugates and finite base changes.Suggested move: Track nonabelian extension constraints rather than pairwise classes, then state an exact recognition theorem generating the module from Gauss–Manin objects.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

The conjecture that every G-operator is of geometric origin remains open. Fresán, Lam, and Qin's 2025 preprint gives counterexamples to a distinct Dwork conjecture about hypergeometric representation; its examples are themselves of geometric origin, so that result neither proves nor disproves the broader Bombieri–Dwork geometric-origin conjecture.

[1]

External progress

What the literature has established

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

2026
Authoritative summaryAndré's current survey restates the motivic and Picard–Fuchs context for G-functions and their conjectural geometric origin.[1]
2025
PreprintFresán, Lam, and Qin give negative answers to Siegel's problem and to a distinct hypergeometric-representation conjecture of Dwork; this is not the broader geometric-origin conjecture staged here.[2]

Mathematical neighborhood

Related results and reusable starting points

Current focusBombieri–Dwork conjecture

Weaker or relaxed formglobally nilpotent and regular-singular G-operators

Known arithmetic properties are necessary features of geometric differential equations but do not by themselves establish geometric origin.

[1]

Related problemhypergeometric representation of G-functions

The preprint constructs geometric-origin rank-two examples that are not algebraic pullbacks of hypergeometric local systems; this narrower negative result must remain distinct from geometric recognition of arbitrary G-operators.

[2]

Formalization opportunities

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

Formalization targetA statement-aligned formalization must fix every ambient field, regularity, genericity, equivalence, and exceptional-set convention appearing in the external statement.
Formalization targetIt must formalize the exact prerequisite geometry, analysis, topology, or arithmetic library needed to express the theorem without replacing it by a weaker proxy.
Formalization targetNo problem-level formal statement or proof was identified in the bounded current Mathlib documentation and public Formal Conjectures repository search.

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 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.

6 standing statements2 proposed statements4 open questions1 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction2 of 82
lemma2 of 82
counterexample2 of 82
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 statementArithmetic differential equations satisfying the G-function bounds all come from geometry.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementG-operator targetintermediate
retained route statementRank-two laboratoryintermediate
retained route statementRecursive-lifting counterexampleintermediate
retained route statementLocal-to-global obstructionintermediate
retained route statementRecognition 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 counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · reported by source
Recorded relationshipThis source-reported counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · 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 failurePromoting first-column or local Cartier success to a full Frobenius extensionreported failure
Research targetProduce a fully normalized geometric calibration example with an independently known Frobenius matrix.open
Research targetConstruct an invertible rank-two semilinear Frobenius matrix with uniform dagger radius and matching local and global extension data.open
Research targetExtend to higher rank and prove a geometric-recognition theorem compatible with conjugates and finite base changes.open
Research targetNormalized weak-link bridgeopen
Narrowed routePromoting first-column or local Cartier success to a full Frobenius extensionThe current work gives separate second-stage, Hodge-source, and global H1 adversarial examples. A calibrated geometric example followed by canonical Galochkin–Cartier comparison, all-order uniform lifting, higher-rank reconstruction, and geometric recognition remains the stated route.

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 bridgeProduce a fully normalized geometric calibration example with an independently known Frobenius matrix.

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.

Continue the mathematics

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Prepared starting pointProduce a fully normalized geometric calibration example with an independently known Frobenius matrix.

Bombieri–Dwork Conjecture for G-functions · ready to start

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Research contextPrepared context for any AI agent

Does every minimal differential equation of a G-function arise from algebraic geometry through Gauss–Manin or Picard–Fuchs constructions?

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

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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
G-functions, motives, and unlikely intersections – old and newsurvey or monograph · Yves André · EMS Press · 2026 · DOI 10.4171/RLM/1075 · accessed Aug 14, 2026
2
On Siegel's problem and Dwork's conjecture for G-functionspreprint · Javier Fresán, Yeuk Hay Joshua Lam, Yichen Qin · arXiv · 2025 · ARXIV 2502.02147 · 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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