{"id":"2652292d-c45c-47e6-ad7b-f5bf864edd7f","arxiv_id":"2505.12521","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In characteristic p, every infinite subset of the prime-to-p Hecke orbit of an ordinary big-monodromy point on a Hodge-type Shimura variety is Zariski dense in the union of connected components.","lead":"This paper proves a positive-characteristic analogue of the André-Pink-Zannier conjecture for ordinary points with big monodromy on Shimura varieties of Hodge type. Infinite subsets of prime-to-p Hecke orbits are shown to be Zariski dense on the whole variety, with applications to Jacobians in isogeny classes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unproved reduction in Section 4 from arbitrary Hecke correspondences fixing V to powers of one h is load-bearing; without it Theorem 2.17 cannot provide rational points on Hecke translates and Tate-linearity does not follow.","rationale":"The reader identified the same weakest assumption: Section 4's transition to a single h and its powers. My reading of the proof confirms that this is the point where the argument's main mechanism, the use of Theorem 2.17 and Proposition 4.2 to obtain a nontrivial Hecke automorphism acting without invariants on the tangent space, is anchored. Without a proof that the infinite sequence of fixing correspondences can be chosen as powers of one semisimple element, the subsequent construction of Weyl-special points and the application of Chai's rigidity theorem do not get off the ground. I do not see an alternative in the manuscript that would supply the same input; the bounded-degree arguments in Proposition 3.9 and the double-coset computation in Lemma 3.10 only control degrees and irreducibility, not the cyclic structure of the stabilizer. This is a serious but potentially repairable gap: if the authors can prove the reduction, or replace Theorem 2.17 by a version valid for arbitrary sequences of correspondences with a common open U, the theorem would go through. Because the claim is plausible and the surrounding architecture is standard, I do not recommend rejection; conditional acceptance pending this justification is appropriate, matching the reader's verdict. I agree with the reader's assessment and do not alter the verdict.","tokens_in":20322,"tokens_out":7011,"duration_ms":73750,"concrete_test":"Attempt to prove the asserted reduction: from the sequence (τ'_j) of Corollary 3.3, construct h∈G(A_f^p) and an infinite subsequence with τ'_j=τ_{h^{n_j}} up to elements of K acting trivially on V. A direct way to test necessity is to work in G=GL_2, K=GL_2(Ẑ^p), and examine the double-coset algebra generated by the Hecke operators T_ℓ for infinitely many ℓ: if a subvariety V can be fixed by T_ℓ and T_m with ℓ≠m but not by any single T_h^n, then Theorem 2.17 cannot be applied and Proposition 4.2 fails. If no such counterexample exists in the relevant Shimura setting, the missing lemma should be written out and checked against Lemma 3.10.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After Corollary 3.3 the proof asserts: 'we may assume that τ_1=τ_h and τ_n=τ_{h^n}' for a single h∈G(A_f,p). Corollary 3.3 only supplies an infinite sequence of distinct prime-to-p Hecke correspondences fixing V_{i,0}; it gives no reason why these lie in the cyclic semigroup generated by one double coset. This matters because Theorem 2.17 is proved for powers h^n of a fixed semisimple element: it produces an open U⊂K such that φ_x∈U implies τ_{h^n}(x) has an F_q-point for every n. Proposition 4.2 then uses that to find y∈V(F_q) lying in infinitely many τ_{h^{n_i}}(x), and hence a nontrivial τ' fixing x and V/x. If the fixing correspondences are not common powers, there is no single U and no such τ'; the proof of Theorem 4.1 (Tate-linearity), and therefore Theorems 2.13, 1.3 and 1.2, is missing its key input. The manuscript contains no lemma bridging Corollary 3.3 to a common power structure, and no argument excluding non-cyclic stabilizers in the Hecke algebra.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a positive-characteristic analogue of the André–Pink–Zannier conjecture for ordinary function-field-valued points with big monodromy on simple Hodge-type Shimura varieties with hyperspecial level at p. The main geometric engine is Theorem 1.3: a generically ordinary curve with big monodromy has the property that the Zariski closure of a union of irreducible components of its prime-to-p Hecke translates is a finite union of connected components of the Shimura variety. From this the authors derive Theorem 1.2 on Zariski closures of infinite subsets of prime-to-p Hecke orbits of ordinary big-monodromy points over F_p(t), an algebraic Hecke-equidistribution statement (Theorem 1.6), and a finiteness result for Jacobians in a prime-to-p isogeny class (Corollary 1.7). The proof proceeds by reducing to maximal monodromy, using degree bounds and finiteness to find Hecke correspondences fixing the limiting subvariety V, proving abundance of Weyl-special points on V, invoking Chai's rigidity theorem to deduce Tate-linearity of V, and then using D'Addezio's parabolicity theorem to conclude that V is a union of connected components. Section 5 spells out the passage from the geometric curve statement to the function-field point statement.","tokens_in":20582,"tokens_out":15586,"duration_ms":152067,"significance":"If the arguments are completed, this would be the first positive-characteristic André–Pink–Zannier-type theorem for Hodge-type Shimura varieties, and it gives a genuinely new algebraic analogue of Hecke equidistribution. The architecture is coherent and ambitious, and the paper makes appropriate use of deep recent inputs: Cadoret–Hui–Tamagawa and Böckle–Gajda–Petersen for monodromy maximality, Chai's rigidity, van Hoften's work on Hecke orbits, and D'Addezio's proof of the parabolicity conjecture. I found no circularity in the main theorems: the self-citations [ST18] and [ST25] are used for auxiliary examples and for an irreducibility argument, not as the substance of Theorem 1.2. The paper also contains several clean, self-contained lemmas, such as Proposition 2.6 and Lemma 3.10, which are useful contributions in their own right. However, the manuscript is not yet complete: a load-bearing reduction in Section 4 is asserted without proof, and a few other steps are under-derived. These issues are local rather than a fundamental flaw in the architecture, but they block acceptance in the current form.","major_comments":[{"comment":"The proof asserts 'we may assume that τ_1=τ_h and τ_n=τ_{h^n}' for a single h∈G(A_f,p). Corollary 3.3 supplies only an infinite sequence of distinct prime-to-p Hecke correspondences τ'_j with τ'_j(V_{i,0})=V_{i,0} for some i; it gives no argument that these correspondences lie in the cyclic semigroup generated by one double coset, nor that they are powers of a common semisimple element. This is load-bearing because Theorem 2.17 is proved for the family {h^n} with a single open subset U: it produces U⊂K such that φ_x∈U implies τ_{h^n}(x) contains an F_q-rational point for every n. Proposition 4.2 and Theorem 4.5 then depend on one such U at a point x∈V. If the fixing correspondences are not common powers, there is no single U, and the construction of the nontrivial correspondence τ' fixing x and V/x collapses. The authors need a lemma that either upgrades Corollary 3.3 to a common-power statement or otherwise provides a finite set of semisimple elements whose powers generate the relevant stabilizer in the Hecke algebra. Without such a lemma, the proof of Theorem 4.1, and therefore of Theorems 2.13, 1.3, and 1.2, is missing a key input.","section":"§4, opening paragraph after Corollary 3.3"},{"comment":"The sentence 'We may also require that the action of Frobenius at x is in U' is not justified by Proposition 4.9 as stated. Proposition 4.9 permits prescribing Frobenius conjugacy classes at finitely many primes ℓ_j∈S_split, with the prescribed elements lying in Im(G_sc(Z/ℓ_j^2)→G(Z/ℓ_j^2)). But U is an open subset of K_ℓ for the fixed auxiliary prime ℓ of Theorem 2.17, and ℓ need not occur among the Weyl-group primes ℓ_w. Even if ℓ is included in the finite set, the authors must prove that U meets Im(G_sc(Z_ℓ)→G(Z_ℓ)) and that the Chebotarev argument can simultaneously impose a class in this intersection at ℓ while imposing the Weyl-torus classes at the ℓ_w. This compatibility is necessary to apply Proposition 4.2 at the Weyl special point, so it is a load-bearing step rather than a mere technicality.","section":"§4.1, proof of Theorem 4.5"},{"comment":"The final paragraph of §5 is too compressed to verify the reduction from the function-field point statement to the curve statement. In particular, from the boundedness of the degrees and fields of definition of the V_{i,0,c} one obtains an infinite subsequence of indices i with V_{i,0,c} equal to a fixed subvariety W; the text instead says 'we find i, j such that V_{i,0,c}=V_{j,0,c}' and then immediately concludes that 'we have a sequence of non-trivial Hecke correspondences which fix the subvariety V_c'. That conclusion requires the same incidence-geometric argument as in Theorem 3.2, showing that V_c⊂σ_i(V_c) for infinitely many distinct Hecke correspondences σ_i. Moreover, the relationship between the original Zariski closure V_F of the points {y_i} and the spread-out families V_{i,0,C} needs to be stated explicitly as a base-change compatibility; as written, the reader cannot check that Tate-linearity of V_c implies the desired conclusion for V_F. I recommend expanding this argument substantially.","section":"§5, proof of Theorem 5.1"}],"minor_comments":[{"comment":"The abstract contains the typo 'characterstic' in the final sentence.","section":"Abstract"},{"comment":"The notation 'had ℓ ∈ T(Qℓ) \\ KT' is awkward; the superscript 'ad' is not explained at first use, and the intended condition should be stated more cleanly, for instance in terms of the image of h in the adjoint group and the maximal compact subgroup of the torus.","section":"Definition 2.8"},{"comment":"The notation 'π_1,et(R)' for a field R is nonstandard; for a field, the étale fundamental group is the absolute Galois group, and the notation should be defined or replaced by Gal(R_s/R) to avoid confusion.","section":"§2.2.3"},{"comment":"In the displayed statement, the modulus 'ℓ k n' appears to be a typo for 'ℓ_j^k'; as written it is not clear whether the Frobenius conjugacy condition is imposed modulo ℓ_j^k for each j. Please correct the notation and also clarify that the number n of primes is separate from the exponent k.","section":"Proposition 4.9"},{"comment":"The proof uses '[ST25, Proposition 5.2]' for the irreducibility of τ_h(Y); since this is an auxiliary and potentially nontrivial input, a brief statement of what Proposition 5.2 provides would help the reader.","section":"Corollary 2.11"},{"comment":"The argument that the action of T_der^x has no trivial subrepresentation is stated in a single sentence; since this is the point where Chai's rigidity hypothesis is verified, a few more details on the Hodge cocharacter and the unipotent representation would improve readability.","section":"§4.2, proof of Theorem 4.1"},{"comment":"The final step cites '[Cha]' (an unpublished preprint) for the rank statement about formal subtori; the precise theorem, with hypotheses, should be stated or quoted, even if the published alternative [vH24] is also referenced.","section":"§4.3, proof of Theorem 2.13"}],"recommendation":"major_revision","confidential_remarks":"The single most important issue is the unproved reduction in Section 4 to a common-power family of Hecke correspondences; this is load-bearing for the Tate-linearity theorem. The paper is well structured and the main results are likely correct in spirit, so I recommend major revision rather than rejection. The authors should also strengthen the final paragraph of Section 5 and make precise the use of Chai's preprint. I did not find circularity or fitted parameters."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the first real characteristic-p analogue of the André–Pink–Zannier conjecture, and the main architecture is sound. The paper deserves a serious referee, but it has two or three places where the proof leaves out steps that a referee will need filled.\n\nWhat is actually new: Theorem 1.2 proves APZ for ordinary function-field points with big monodromy on simple Hodge-type Shimura varieties; Theorem 1.3 is a geometric Hecke-stability statement; Theorem 4.5 gives a characteristic-p proof of abundance of Weyl special points; and Theorem 1.6 is a natural algebraic Hecke equidistribution result. The overall strategy—reduce to maximal monodromy, prove Hecke stability, find Weyl special points, apply Chai rigidity, then D’Addezio’s parabolicity—is coherent and matches standard practice. The self-citations are used as context or auxiliary input, not as the content of the main theorems.\n\nThe soft spots are real but not fatal. The largest is in Section 4: after Corollary 3.3, the text says 'we may assume that τ_1=τ_h and τ_n=τ_{h^n}' with no supporting argument. Corollary 3.3 only gives an infinite sequence of distinct Hecke correspondences fixing V. The leap to a common power structure is probably true—an infinite set of double cosets in the compact Hecke algebra must contain a non-torsion element, whose powers then work—but the paper should state and prove that lemma. As written, this is a genuine gap, though a fillable one. The stress-test note is correct to flag it, but I do not think it sinks the paper.\n\nA second issue is Corollary 1.7: as stated, it says that for any proper subvariety D of SK(G,X), the Hecke orbit meets D in only finitely many points. If SK(G,X) has several connected components, D could contain one connected component, and then the claim is false. The fix is to require D to contain no connected component of SK(G,X), or to work on a single connected component throughout. This affects only the corollary, not the main theorems.\n\nMentioning the smaller things: Corollary 2.15’s proof is terse but checks out, and Theorem 3.2’s indexing could be clearer but the argument is standard. The reliance on deep cited theorems is normal for this area.\n\nMy recommendation: send it to a serious referee. The main result is important and the proof is credible, but the Section 4 reduction and the Corollary 1.7 statement need to be addressed before publication.","headline":"A serious and largely sound first positive-characteristic APZ theorem, but with a few proof transitions that need explicit lemmas before the paper is fully convincing.","tokens_in":21134,"tokens_out":6351,"would_cite":true,"duration_ms":63719,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G18","14G35","14K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a positive-characteristic analogue of the André–Pink–Zannier conjecture: on a simple Hodge-type Shimura variety with hyperspecial level at $p$, any infinite subset of the prime-to-$p$ Hecke orbit of an ordinary point over…","keywords":["André–Pink–Zannier conjecture","characteristic p","Shimura variety of Hodge type","Hecke correspondence","ordinary locus","big monodromy","Weyl special point","algebraic equidistribution"],"falsifier":"Find an ordinary point $x \\in SK(G,X)(\\mathbb{F}_p(t))$ with big monodromy and an infinite sequence of distinct prime-to-$p$ Hecke translates of $x$ all lying in a single proper subvariety $D \\subset SK(G,X)$; for instance, for $g \\ge 4$, an ordinary big-monodromy abelian variety over $\\mathbb{F}_p(t)$ whose prime-to-$p$ isogeny class contains infinitely many Jacobians. Such an example would contradict the finiteness corollary and therefore the main theorem.","tokens_in":20126,"feed_emoji":"🧮","tokens_out":14358,"duration_ms":126636,"temperature":0.7,"pith_summary":"The paper proves a positive-characteristic analogue of the André–Pink–Zannier conjecture, a central statement about unlikely intersections on Shimura varieties. For a simple Shimura variety of Hodge type with hyperspecial level at $p$, it shows that any infinite subset of the prime-to-$p$ Hecke orbit of an ordinary point over $\\mathbb{F}_p(t)$ with big monodromy is Zariski dense: its closure is a finite union of connected components. The same argument yields an algebraic form of Hecke equidistribution for generically ordinary curves with big monodromy. A direct consequence is a finiteness theorem: only finitely many points of such a Hecke orbit can lie on a fixed proper subvariety, so for $g \\ge 4$ only finitely many Jacobians occur in a given prime-to-$p$ isogeny class of ordinary abelian varieties. This matters because characteristic $p$ has no archimedean equidistribution, and the authors replace it with a purely algebraic mechanism based on Hecke-stable subvarieties, Weyl special points, and Tate-linear subvarieties.","feed_headline":"Infinite Hecke orbits of ordinary points are Zariski dense in char p","feed_subtitle":"It proves the function-field analogue of the André–Pink–Zannier conjecture for Hodge-type Shimura varieties.","key_machinery":"The argument replaces the missing archimedean equidistribution with a chain of algebraic rigidity statements. After passing to a subvariety with maximal geometric prime-to-$p$ monodromy, degree arithmetic on the Baily–Borel compactification—the identity $\\deg(W_{i,0}) = n_1 \\deg(V) = n_2 \\deg(V_{i,0})$ together with a lower bound $\\kappa \\deg(\\tau_i) \\le n_1, n_2$—forces the closure $V$ to be fixed by an infinite sequence of distinct Hecke correspondences. The next step shows that ordinary Weyl special points are abundant on $V$: these are points whose associated $\\mathbb{Q}$-torus has a Galois action on its character lattice containing the full Weyl group, and they are produced by a Frobenius-conjugacy argument from tori in split reductions. At such a point, a rigidity theorem for $p$-divisible formal groups shows that a formal subscheme fixed by a nontrivial automorphism with no invariant tangent directions is a formal subgroup; since $V$ is fixed by such an automorphism, $V$ must be Tate-linear. Finally, the parabolicity theorem for F-isocrystals identifies the non-overconvergent $p$-adic monodromy of the slope filtration with a parabolic subgroup, and the formal subgroup containing $V$ has full dimension, so $V$ is a union of connected components.","core_discovery":"Let $SK(G,X)$ be a simple Shimura variety of Hodge type embedded in a moduli space of principally polarized abelian varieties, with hyperspecial level at $p$, and suppose its special fiber meets the ordinary locus. The paper proves that if $x \\in SK(G,X)(\\mathbb{F}_p(t))$ is ordinary and has big monodromy, then every infinite subset $\\Sigma$ of the prime-to-$p$ Hecke orbit of $x$ has Zariski closure equal to a finite union of connected components of $SK(G,X)$. The geometric engine is a companion statement for curves: an irreducible curve $C$ with big monodromy that meets the ordinary locus is taken by any sequence of distinct prime-to-$p$ Hecke correspondences with degrees tending to infinity to curves $C_i$ whose union is Zariski dense, which forces the Hecke orbit of $C$ to equidistribute in the algebraic sense. A corollary is that a proper subvariety $D$ of $SK(G,X)$ contains only finitely many points of the prime-to-$p$ Hecke orbit of such an $x$; in particular, an ordinary $g$-dimensional abelian variety over $\\mathbb{F}_p(t)$ with big monodromy is prime-to-$p$ isogenous to only finitely many Jacobians for $g \\ge 4$.","pith_inferences":["The authors state that the big-monodromy hypothesis should be removable; if so, the Zariski-density conclusion would hold for every ordinary point of a Hodge-type Shimura variety over a function field, leaving the restriction to $\\mathbb{F}_p(t)$ (rather than arbitrary fields) as the essential boundary.","The same degree-bound and rigidity mechanism may apply to Hecke-stable strata beyond the ordinary locus, where the conclusion should be phrased as density in the smallest Hecke-invariant union of central leaves because Newton strata themselves are Hecke-invariant.","A natural strengthening is whether an ordinary big-monodromy Jacobian over a function field is prime-to-$p$ isogenous to no other Jacobian; the paper proves only finiteness, not nonexistence.","Because the proof is algebraic rather than analytic, it gives a template for algebraic equidistribution on other moduli problems with a usable theory of ordinary points and canonical lifts, such as K3 surfaces."],"forward_implications":["No infinite subset of the prime-to-$p$ Hecke orbit of an ordinary big-monodromy point can be Zariski dense in a proper subvariety; its closure is an entire union of connected components.","For a generically ordinary curve with big monodromy, every sequence of Hecke correspondences with degree tending to infinity makes the translated curves equidistribute algebraically: the minimal degree of a hypersurface containing them tends to infinity.","Any proper subvariety $D$ of $SK(G,X)$ defined over $\\mathbb{F}_p(t)$ contains only finitely many points of the prime-to-$p$ Hecke orbit of such an $x$.","For $g \\ge 4$, an ordinary $g$-dimensional abelian variety over $\\mathbb{F}_p(t)$ with big monodromy is prime-to-$p$ isogenous to only finitely many Jacobians."],"supporting_citations":[{"why":"Constructs the integral canonical model over the local ring at $p$, so the mod $p$ special fiber of the Hodge-type Shimura variety is well defined.","marker":"[Kis10]"},{"why":"Shows geometric $\\ell$-adic monodromy of a subvariety equals that of the ambient Shimura variety for all but finitely many $\\ell$, allowing reduction to maximal monodromy.","marker":"[CHT17]"},{"why":"Gives independence of $\\ell$-adic representations of geometric Galois groups, used to pass from big to maximal prime-to-$p$ monodromy and in the Frobenius-conjugacy step.","marker":"[BGP18]"},{"why":"Defines Weyl special points and proves their abundance in characteristic zero; the paper adapts the construction to characteristic $p$ via split reductions.","marker":"[CO12]"},{"why":"Provides the rigidity result that a formal subscheme fixed by a nontrivial automorphism with no invariants on the tangent space is a formal subgroup.","marker":"[Cha08]"},{"why":"Supplies canonical coordinates, Tate-linear subvarieties, and the statement that the smallest formal subtorus containing $V/x$ has rank equal to the unipotent radical of the $p$-adic monodromy.","marker":"[Cha]"},{"why":"Proves the parabolicity conjecture for F-isocrystals, identifying the non-overconvergent $p$-adic monodromy $P$ as the parabolic associated to the slope filtration.","marker":"[D'A23]"},{"why":"Proves the ordinary Hecke orbit conjecture for Hodge-type Shimura varieties, used in the final step that $V$ is a union of connected components.","marker":"[vH24]"},{"why":"Establishes the function-field abelian varieties not isogenous to Jacobians and gives the monodromy argument used to show the function-field closure has big monodromy.","marker":"[ST25]"},{"why":"Provides the Baily–Borel compactification and extension of the Hodge line bundle used to define degrees of subvarieties and prove the degree bounds.","marker":"[Per19]"}],"fun_headline_variants":["Char p André–Pink–Zannier: Hecke orbits dense","Hecke orbits of ordinary points: Zariski dense in char p","APZ conjecture for char p Shimura varieties of Hodge type","Infinite Hecke orbits dense in characteristic p"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is the assertion that the infinite sequence of distinct Hecke correspondences fixing the subvariety can be chosen as powers of a single nontrivial correspondence, so that it acts on the tangent space at a fixed Weyl special point with no nonzero fixed vectors; the paper does not construct such a correspondence or rule out the possibility that all fixing correspondences are of finite order.","fun_headline_variants_meta":{"raw":{"variants":["Char p André–Pink–Zannier: Hecke orbits dense","Hecke orbits of ordinary points: Zariski dense in char p","APZ conjecture for char p Shimura varieties of Hodge type","Infinite Hecke orbits dense in characteristic p"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"total_tokens":1362,"prompt_tokens":935,"completion_tokens":427,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":353}},"tokens_in":551,"tokens_out":427,"duration_ms":4215,"temperature":1.0,"reasoning_tokens":353,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:34:06.278566+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find an ordinary point $x \\in SK(G,X)(\\mathbb{F}_p(t))$ with big monodromy and an infinite sequence of distinct prime-to-$p$ Hecke translates of $x$ all lying in a single proper subvariety $D \\subset SK(G,X)$; for instance, for $g \\ge 4$, an ordinary big-monodromy abelian variety over $\\mathbb{F}_p(t)$ whose prime-to-$p$ isogeny class contains infinitely many Jacobians. Such an example would contradict the finiteness corollary and therefore the main theorem.","supporting_citations":[],"review_version":1}