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Transport of Zariski density in compatible collections of $G$-representations

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Pith's one-line read A compatible collection of G-representations with full algebraic monodromy at a finite G-good set of primes has full monodromy on a set of primes of Dirichlet density 1, with applications to non-Abelian Shimura varieties.

desk verdict A credible new propagation theorem for Zariski density in compatible systems, with two small repairable gaps in the proof. read the letter →

arxiv 2411.08259 v1 pith:IP6CXVWF submitted 2024-11-13 math.NT

Jake Huryn , Yifei Zhang This is my paper · ORCID
classification math.NT
keywords mathbfconnecteddensityfinitemathcalmathrmtypezariski-dense
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The second part of the paper converts this into a statement about specializations. Hilbert's irreducibility theorem is used to find many points on a scheme whose restricted representations inherit the full image. Combining this with recent work of Klevdal and Patrikis on canonical local systems on adjoint Shimura varieties, the authors obtain infinitely many closed points on Shimura varieties of non-Abelian type, such as those of type E6 and E7, where the monodromy is full for almost all primes. Earlier strong results existed only for Abelian type. The main reservation in the proof is a small omission: the primes used in the construction must avoid the finite set where the group is not quasisplit, and the paper does not say this explicitly.
Extended reading notes

Core claim

Theorem A (Theorem 3.2.1): Let {ρ_l : Γ -> G(Q_l)}_{l in L} be an abstract compatible collection with L of Dirichlet density 1. If there is a G-good finite set R subset of L with M_l = G_{Q_l} for each l in R, then {l in L : M_l = G_{Q_l}} has Dirichlet density 1. If true, full algebraic monodromy propagates from a finite, diagram-seeing set of primes to almost all primes, giving a partial positive answer to Question 1.1.3.

Load-bearing premise

In Step 2 and Step 3 of the proof of Theorem A, the authors select primes l_{i,omega,xi} that are completely split in E|Q and lie in L, and then use Lemma 2.1.5 (from [LP92]) to construct maximal tori with prescribed Frobenius classes. Lemma 2.1.5 requires G_{Q_l} to be quasisplit. The text does not explicitly require these chosen primes to avoid the finite set of primes where G_{Q_l} is not quasisplit; since only finitely many primes are bad, the gap is readily fixable, but as written the proof silently depends on this selection.

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Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper studies abstract compatible collections of continuous representations ρ_ℓ: Γ → G(Q_ℓ) of a profinite group Γ into a connected reductive group G over Q, indexed by a set L of primes of Dirichlet density 1. The main result, Theorem A (Theorem 3.2.1), states that if there is a finite set R ⊆ L, one prime for each Frobenius class in the Dynkin-diagram splitting field E of G, at which the Zariski closure of the image is all of G and G is quasisplit, then the set of ℓ ∈ L at which the algebraic monodromy group M_ℓ equals G_{Q_ℓ} has Dirichlet density 1. The proof reduces to the adjoint case, uses Larsen–Pink torus constructions and a lemma showing that a reductive subgroup containing maximal tori in every relevant Frobenius class must be the whole group, and derives a Chebotarev-based density estimate. Theorem B (Theorem 4.2.2) uses a Hilbert irreducibility result for profinite groups to produce many closed points x of a scheme X for which the same density-one conclusion holds for the specialized representations ρ_{ℓ,x}; this is applied to canonical local systems on Shimura varieties not of Abelian type, via work of Klevdal–Patrikis.

Significance. If the main theorem is correct, it gives a substantial partial positive answer to Question 1.1.3, showing that full algebraic monodromy at a finite, diagram-seeing set of primes propagates to a Dirichlet-density-one set of primes in a general abstract compatibility framework. This goes beyond the rank and component-group independence results of Serre and complements the Larsen–Pink theory by focusing on the maximal case M_ℓ = G. The application to non-Abelian Shimura varieties, using the recent compatibility result [KP24], is a valuable and timely novelty: it yields compatible systems with Zariski-dense image for adjoint groups of types E_6 and E_7 that are not of Abelian type, where motivic methods are currently unavailable. The paper is carefully written, with a clear reduction to adjoint semisimple groups, a transparent axiomatization of compatible collections, and extensive credit to [LP92]. The proof is largely self-contained modulo the quoted external results. The two technical gaps identified below are localized and appear repairable without changing the main conclusions.

major comments (2)
  1. [§3.2.1, Eq. (3.2.1.2)] The displayed inequality is not correct as written. The second inequality in (3.2.1.2) asserts [F_i:E] ≤ #W(Ψ_0)·#[Ω], but the preceding construction forces much larger degrees. For each fixed (i,ω), the Frobenius elements at the auxiliary primes ℓ_{i,ω,ξ} lie in Gal(F_{T_{i,ω}}/E) and, by (3.2.1.1), their images under the injective map φ_{T_{i,ω}} meet every conjugacy class of W(Ψ_0); hence Gal(F_{T_{i,ω}}/E) ≅ W(Ψ_0). Since F_i is the composite of the #[Ω] linearly disjoint fields F_{T_{i,ω}} for fixed i, one has [F_i:E] = (#W(Ψ_0))^{#[Ω]}, which is generally much larger than #W(Ψ_0)·#[Ω]. The density conclusion of Theorem A is nevertheless salvageable: replacing the constant #W(Ψ_0)·#[Ω] by (#W(Ψ_0))^{#[Ω]} in the final bound gives (1 - (#W(Ψ_0))^{-#[Ω]})^n, which still tends to 0 as n → ∞. The proof therefore needs a corrected estimate, but the main claim survives.
  2. [§3.2.1, Steps 2 and 3] The proof invokes Lemma 2.1.5 (from [LP92]) to construct maximal tori with prescribed Frobenius classes, but Lemma 2.1.5 requires the ambient group G_{Q_ℓ} to be quasisplit. In Step 2, the prime ℓ_{1} is not explicitly chosen to be an element of the finite set R with trivial Frobenius class in Gal(E|Q), which is needed both for M_{ℓ_1} = G and for G_{Q_{ℓ_1}} to be quasisplit. In Step 3, the primes ℓ_{i,ω,ξ} are required only to be completely split in E|Q and to satisfy M_{ℓ_{i,ω,ξ}} = G; Lemma 2.1.5 is then applied at these primes without excluding the finitely many primes where G_{Q_ℓ} is not quasisplit. Since only finitely many primes are bad, the argument is readily repairable by explicitly requiring all these selections to avoid the finite bad set, but as written the proof silently depends on this extra condition.
minor comments (3)
  1. [§3.1.1, Definition 3.1.1(a)] The notation 'ℓ∈L /integerdivideS_f' appears garbled; it should read ℓ∉S_f (or ℓ∈L\S_f). The same symbol recurs later, for example in the proof of Lemma 3.1.2, and should be normalized throughout.
  2. [§3.2.1, Step 3, linear-disjointness claim] The proof that F_{T_{i,ω}} ∩ F^{(i,ω)} = E concludes 'again since ξ is arbitrary', but the underlying group-theoretic fact is not spelled out: because the kernel of the restriction map is normal, if it contains one element from every conjugacy class of Gal(F_{T_{i,ω}}/E), then it contains every conjugacy class and hence the whole group. A short explicit sentence would improve readability.
  3. [§1.2.1 and §3.2.1] The terminology 'G-good' is introduced in Definition 1.2.1 with only conditions (a)–(c), while Theorem 3.2.1 states its hypotheses directly with an additional full-monodromy condition (d). The relationship between the two formulations is clear, but the paper would be easier to follow if the theorem explicitly said 'let R be a G-good set such that M_ℓ = G for each ℓ∈R'.

Circularity Check

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No circularity: Theorem A is proved from abstract compatibility and external group-theoretic results, with the Shimura application relying on external compatibility results.

full rationale

Theorem A (Theorem 3.2.1) assumes a compatible collection and a finite G-good set R with full monodromy, and concludes a Dirichlet-density-one set of full monodromy. The proof reduces to the adjoint maximal-rank case, uses Lemma 3.1.2 to transport torus classes from finitely many prescribed primes, and then uses Lemma 2.3.5 with maximal tori constructed from [LP92, Lemma 3.6]; none of these steps uses the desired density-one conclusion as an input. The linear disjointness claim in Step 3 is argued from the explicitly prescribed Frobenius values in (3.2.1.1), not from the conclusion. The cited external results [LP92], [Ser13a], [Ser65], and [Ser89] are independent support. The application Corollary 1.3.2 depends on [KP24, Theorem 1.3] for compatibility of canonical local systems; this is an external input and is not the conclusion being transported, so the involvement of the authors' advisor is not a load-bearing self-citation. A reviewer concern is that the displayed density bound (3.2.1.2) appears to underestimate [F_i:E] (which may be (#W)^(#Omega)), but this is a proof-correctness issue, not a circularity, and the argument is likely repairable. No definitional, fitted-input, or self-citation circularity was found.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central theorem rests on the abstract compatibility notion and on several external results from [LP92], [Ser13a], [Ser89], and the Stacks project; none of these are fit to the target conclusion. No free parameters are introduced, and no new entities are postulated. The auxiliary objects, such as the fields F_{T_{i,omega}} and tori, are constructed explicitly inside the proof.

assumptions (8)
  • domain assumption Compatibility of the collection relative to a dense subset F (Definition 3.1.1): for each f in F, the semisimple parts rho_l(f)_ss are conjugate to a fixed rational conjugacy class away from a finite set S_f, and the sets F(l_1,...,l_n) are dense in Gamma.
    This is the definition of abstract compatible collection used in Theorems A and B; it is stronger than the usual GL_n trace compatibility but is the natural formulation for G-valued representations.
  • standard math Larsen-Pink Lemma 3.6 [LP92, Lemma 3.6]: over a field K, every W-conjugacy class of lifts of phi_G is realized by a maximal torus of a quasisplit group G.
    Used to produce tori with prescribed Frobenius classes at the chosen primes in Steps 2 and 3 of Theorem 3.2.1.
  • standard math Larsen-Pink Proposition 7.3 [LP92]: in a compatible system, a dense subset F contains elements whose semisimple parts specialize into prescribed maximal tori at finitely many primes.
    Used in Lemma 3.1.2 to move torus-data from a finite set of primes to almost all primes.
  • standard math Serre's rank-independence theorem [Ser13a, §3]: the rank and component group of the algebraic monodromy M_l are independent of l for compatible collections.
    Used in Step 1 of Theorem A to reduce to the case where every M_l is reductive of maximal rank.
  • standard math Chebotarev density theorem for number fields.
    Provides the density-one and density-zero estimates for primes by Frobenius class in the fields E and F_i.
  • standard math Serre's Hilbert irreducibility theorem for profinite groups with open Frattini subgroup [Ser89, §10.6].
    Used in Theorem B to produce infinitely many specialization points with prescribed image in a product of p-adic groups.
  • standard math Stacks project facts: pi_1(U) to pi_1(X) is surjective for open U, and quotients Y/Sigma of normal schemes are irreducible.
    Used in Proposition 4.1.1 to reduce to affine X and to apply Hilbert irreducibility.
  • domain assumption Compatibility and integral spreading of canonical local systems on adjoint Shimura varieties [KP24, Theorem 1.3].
    External input used only in Corollary 1.3.2, not in the proof of Theorem A.

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Pith. "Pith review of Transport of Zariski density in compatible collections of $G$-representations." pith.science (2026). https://pith.science/paper/IP6CXVWF

@misc{pith2026241108259,
  author       = {Pith},
  title        = {Pith review of: Transport of Zariski density in compatible collections of $G$-representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IP6CXVWF}},
  note         = {Machine review of arXiv:2411.08259}
}
abstract

Let $X$ be a connected normal scheme of finite type over $\mathbf{Z}$, let $G$ be a connected reductive group over $\mathbf{Q}$, and let $\{\rho_\ell\colon\pi_1(X[1/\ell])\to G(\mathbf{Q}_\ell)\}_\ell$ be a Frobenius-compatible collection of continuous homomorphisms indexed by the primes. Assume $\mathrm{Img}(\rho_\ell)$ is Zariski-dense in $G_{\mathbf{Q}_\ell}$ for all $\ell$ in a nonempty finite set $\mathcal{R}$. We prove that, under certain hypotheses on $\mathcal{R}$ (depending only on $G$), $\mathrm{Img}(\rho_\ell)$ is Zariski-dense in $G_{\mathbf{Q}_\ell}$ for all $\ell$ in a set of Dirichlet density $1$. As an application, we combine this result with a version of Hilbert's irreducibility theorem and recent work of Klevdal--Patrikis to obtain new information about the "canonical" local systems attached to Shimura varieties not of Abelian type.

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