Arithmetic geometry · modular forms · Sato–Tate distributions

Generalized Sato–Tate for Cubic GL₂-Type Abelian Threefolds

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For a generic non-CM modular abelian threefold whose coefficient field has three real embeddings, do the three normalized Frobenius angles vary jointly and independently according to the product Sato–Tate measure?

{g_{p}}_{p} ? ⟶ Haar (SU (2)^{3})

Known results and sources

Three coefficient embeddings produce three normalized Frobenius angles; the open target asks for their joint product-Haar distribution in SU(2)³.

Research problem

Exact mathematical statement

Let $f$ be a normalized weight-two newform with trivial nebentype and totally real cubic coefficient field $E$ , assume $f$ is non-CM with no nontrivial inner twists, and let $A_f/\mathbf Q$ be the associated absolutely simple abelian threefold with $\operatorname{End}_{\overline{\mathbf Q}}^0(A_f)=E$ and connected derived monodromy $\operatorname{SL}_2^3$ . For a good prime $p$ , write $\theta_{1,p},\theta_{2,p},\theta_{3,p}$ for the normalized Frobenius angles attached to the three real embeddings of $E$ . The target asks whether, for every continuous $F:[0,\pi]^3\to\mathbf C$ ,

\lim_{X\to\infty}\frac{1}{\pi(X)}\sum_{\substack{p\le X\\p\nmid N}}F(\theta_{1,p},\theta_{2,p},\theta_{3,p})=\int_{[0,\pi]^3}F(\theta_1,\theta_2,\theta_3)\prod_{i=1}^3\frac{2}{\pi}\sin^2\theta_i\,d\theta_i.

Equivalently, the normalized Frobenius classes should be Haar-equidistributed in $\operatorname{SU}(2)^3$ . The governing source says that no full proof is claimed.

Problem infographic

Problem at a glance

A landscape scientific diagram shows one cubic-coefficient modular abelian threefold, three Frobenius angle coordinates, and a target SU(2) cubed product space whose sample points have not yet filled the expected measure. — The exact question compares prime-indexed triples of normalized Frobenius angles with the product of three SU(2) Sato–Tate measures; the packet's automorphic bridge is still conditional.

Current mathematical picture

Where work on Generalized Sato–Tate for Cubic GL₂-Type Abelian Threefolds stands

Open problem

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

Useful failurePre-supplying a faithful Hecke action from the abstract four-branch order

An abstract four-branch order and one-prime coefficient reconstruction cannot be treated as an already faithful Hecke action; doing so assumes the mixed branch that the crossing calculation is meant to establish. Construct a real horizontal automorphic object carrying the needed tensor trace or characteristic-coefficient operators, prove generic and integral factorization with torsion guards, establish independent mixed-branch survival, and extract the exact classical automorphic parameters uniformly across the sparse family.

Route status · Narrowed route

Main reductionCharacter Euler products control equidistribution

The source reduces product-Haar equidistribution to analytic continuation, holomorphy, and nonvanishing of every nontrivial SU(2)³ character Euler product at the boundary.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeConstruct the tensor trace or characteristic-coefficient operators on an actual horizontal automorphic object.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

Generalized Sato–Tate for Cubic GL₂-Type Abelian Threefolds in numbers

10.5kretained lines of mathematical investigation10,467 in the current working snapshot

Argument development: 9,042 · 86%
Explored or eliminated routes: 171 · 2%
Computational analysis: 254 · 2%
Open obligations: 250 · 2%
Definitions and setup: 750 · 7%

6selected 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.

Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Construct the tensor trace or characteristic-coefficient operators on an actual horizontal automorphic object.

Suggested move: For the (1,3) calibration, specify the automorphic category and its integral lattice, derive the tensor pseudotrace or characteristic coefficients from a universal determinant, ambient Hecke polynomial, Cayley–Hamilton algebra, or known commuting Hecke operators, and audit every normalization.

What would count as progress

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

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Other routes

1 recorded

Narrowed routePre-supplying a faithful Hecke action from the abstract four-branch order

Route status · Narrowed route

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Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Construct the tensor trace or characteristic-coefficient operators on an actual horizontal automorphic object.Suggested move: For the (1,3) calibration, specify the automorphic category and its integral lattice, derive the tensor pseudotrace or characteristic coefficients from a universal determinant, ambient Hecke polynomial, Cayley–Hamilton algebra, or known commuting Hecke operators, and audit every normalization.

Ready to work on

Prove generic and integral factorization and a noncircular lattice-sensitive mixed-branch survival certificate.Suggested move: Show the characteristic-zero action factors semisimply through the four-branch order, apply the integral descent theorem with torsion and base-change guards, then compute an independent residual crossing, normalized branch trace, or Lefschetz selector that does not assume the desired branch.

Ready to work on

Extract exact classical automorphic parameters and establish analytic adequacy uniformly over all required sparse pairs.Suggested move: After branch survival, prove local–global compatibility, determinant, polarization, ramified types, descent, cuspidality or controlled isobaric structure, boundary holomorphy and nonvanishing, and uniform collision-prime and lattice bounds as the exponents vary.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen problem

The exact joint SU(2)^3 Frobenius-equidistribution statement for the specified generic cubic-GL2-type modular abelian threefold was not identified as proved in this scoped current-source review. Single-form Sato–Tate, a two-form joint theorem, certain potentially-GL2-type cases, and the group classification for abelian threefolds are substantial neighboring results, but their hypotheses and dimensions do not silently close the exact triple-character target.

[1][2][3]

External progress

What the literature has established

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

2023
PreprintFité, Kedlaya, and Sutherland classified 410 possible Sato–Tate groups for abelian threefolds, realized all maximal groups, and computed moments with numerical checks. Classification and realization do not by themselves establish the exact equidistribution theorem here.[3]
2021
PreprintFité and Guitart introduced Tate-module tensor decompositions and proved Sato–Tate in certain potentially-GL2-type cases over totally real fields; the abstract does not identify those cases with this exact cubic joint target.[2]
2020
Peer reviewedThorner proved effective Sato–Tate for holomorphic cuspidal newforms and an unconditional joint distribution for two twist-inequivalent newforms, a neighboring two-coordinate theorem.[4]
2010
PreprintBarnet-Lamb, Gee, and Geraghty proved the natural Sato–Tate conjecture for non-CM regular algebraic cuspidal GL2 representations over totally real fields, supplying the single-coordinate modular result but not the exact three-coordinate joint theorem.[1]

Mathematical neighborhood

Related results and reusable starting points

Current focusGeneralized Sato–Tate for Cubic GL₂-Type Abelian Threefolds

Solved special caseSingle-coordinate Sato–Tate for Hilbert modular forms

The non-CM single modular-form Sato–Tate theorem controls each coordinate separately but does not automatically give independence of all three coefficient-conjugate coordinates.

[1]

Solved special caseJoint Sato–Tate for two newforms

The effective joint theorem handles two twist-inequivalent newforms. The exact target requires a three-coordinate coefficient-conjugate joint distribution and additional triple tensor sectors.

[4]

Related problemCertain potentially-GL2-type Sato–Tate cases

Tensor decompositions for certain potentially-GL2-type abelian varieties are structurally relevant, but their theorem scope must be matched hypothesis by hypothesis before application to a totally real cubic coefficient field.

[2]

Related problemSato–Tate group classification for abelian threefolds

The classification of possible abelian-threefold Sato–Tate groups supplies group-theoretic context and moments without proving this specific family's prime equidistribution.

[3]

Formalization opportunities

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

Formalization targetA statement-aligned formal model of modular abelian threefolds of cubic GL2 type, their three coefficient-conjugate compatible systems, and normalized Frobenius classes.
Formalization targetFormal representation theory for SU(2)^3 characters and the exact reduction of product-Haar equidistribution to partial Euler-product boundary conditions.
Formalization targetMachine-checked interfaces for the symmetric-power, tensor-transfer, Rankin–Selberg, automorphy, classicality, and nonvanishing inputs with exact fields, local types, and normalizations.

Research-record corrections

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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

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Detailed research inventory

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4 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role6 selected mapped statements

theorem candidate1 of 61
reduction2 of 62
lemma2 of 62
negative result1 of 61

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 statementFor a generic non-CM modular abelian threefold whose coefficient field has three real embeddings, do the three normalized Frobenius angles vary jointly and independently according to the product Sato–Tate measure?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementCharacter Euler products control equidistributionintermediate
retained route statementCoefficient-collision algebra is source-reported completeintermediate
retained route statementAbstract order cannot pre-supply the Hecke imageintermediate
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
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 failurePre-supplying a faithful Hecke action from the abstract four-branch orderreported failure
Research targetConstruct the tensor trace or characteristic-coefficient operators on an actual horizontal automorphic object.open
Research targetProve generic and integral factorization and a noncircular lattice-sensitive mixed-branch survival certificate.open
Research targetExtract exact classical automorphic parameters and establish analytic adequacy uniformly over all required sparse pairs.open
Research targetRestricted SU(2)³ target remains opensuperseded
Research targetTensor operators need genuine provenancesuperseded
Research targetClassical extraction and uniformity remain global gatessuperseded
ComputationSource-reported finite rank counts and bottom-padding certificates at named primes, selected symbolic and finite-range Hasse/Wronskian checks, retained verification outputs, and v18 guard and integrity checks.The governing source reports the listed counts, certificates, checks, and retained outputs and describes what its v18 scripts check. ProofAtlas did not execute those scripts or independently reproduce any result; these reports do not prove the global Sato–Tate target. · reported unreproduced
Narrowed routePre-supplying a faithful Hecke action from the abstract four-branch orderAn abstract four-branch order and one-prime coefficient reconstruction cannot be treated as an already faithful Hecke action; doing so assumes the mixed branch that the crossing calculation is meant to establish. Construct a real horizontal automorphic object carrying the needed tensor trace or characteristic-coefficient operators, prove generic and integral factorization with torsion guards, establish independent mixed-branch survival, and extract the exact classical automorphic parameters uniformly across the sparse family.

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Priority open bridgeConstruct the tensor trace or characteristic-coefficient operators on an actual horizontal automorphic object.

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

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Prepared starting pointConstruct the tensor trace or characteristic-coefficient operators on an actual horizontal automorphic object.

Generalized Sato–Tate for Cubic GL₂-Type Abelian Threefolds · ready to start

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Sources and references4 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
The Sato-Tate conjecture for Hilbert modular formspreprint · Thomas Barnet-Lamb, Toby Gee, David Geraghty · arXiv; final version announced for Journal of the AMS · 2010-11-04 · ARXIV 0912.1054 · accessed Aug 14, 2026
2
Tate module tensor decompositions and the Sato-Tate conjecture for certain abelian varieties potentially of GL2-typepreprint · Francesc Fité, Xavier Guitart · arXiv; final version announced for Mathematische Zeitschrift · 2021-11-04 · ARXIV 1909.11712 · accessed Aug 14, 2026
3
Sato-Tate groups of abelian threefoldspreprint · Francesc Fité, Kiran S. Kedlaya, Andrew V. Sutherland · arXiv; final version announced for Memoirs of the AMS · 2023-07-17 · ARXIV 2106.13759 · accessed Aug 14, 2026
4
Effective forms of the Sato–Tate conjecturepeer reviewed result · Jesse Thorner · Research in the Mathematical Sciences · 2021 · ARXIV 2002.10450 · DOI 10.1007/s40687-020-00234-3 · accessed Aug 14, 2026

Important qualifications

This workspace title identifies a narrow cubic-coefficient joint-distribution target, not the unrestricted generalized Sato–Tate conjecture.
The scoped search found no primary source establishing the exact three-coordinate SU(2)^3 target under the current work's full generic hypotheses; that bounded negative finding does not prove global nonexistence.
A single-form Sato–Tate theorem, a two-form joint distribution, a classification of possible threefold Sato–Tate groups, or a theorem for certain potentially-GL2-type varieties is not silently identified with this exact target.
No submitted URL was fetched, and the current work's internal literature assertions and probability estimate were not used as external-status evidence.
No statement-aligned formalization, checked proof, reusable dataset, or independently reproduced computation was identified in this scoped collection.

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