{"id":"e1dc7490-b703-4244-8065-e8ee7f4c84de","arxiv_id":"2411.16643","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Hecke orbits on the Picard group of definite Shimura curves over Fq(t) equidistribute toward a natural weighted measure, including the supersingular Drinfeld module case.","lead":"This paper proves that Hecke operators move divisors on Shimura curves over function fields so that, after many steps, the divisors spread uniformly over the curve's Picard group with a natural weighting. It uses automorphic forms over function fields, and as a corollary shows the same equidistribution for supersingular Drinfeld modules of rank 2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.3 is the sole bridge from the cusp-form estimate at m_{n0} to the ratio at the full ideal m; its proof is omitted and the cited 'Proposition II.1' is unlocated, so the convergence of the original Hecke orbit is not fully demonstrated.","rationale":"The reader's weakest-assumption analysis already identifies Proposition 2.3 as the load-bearing gap, and my independent reading reaches the same conclusion. The strategy of the paper is otherwise coherent: Brandt matrices encode Hecke movement, the Fourier expansion of Θ_{ij} and E_{n0} gives the cusp form g_{ij}, and the Deligne–Ramanujan bound (3.4) supplies the required decay once the ratio is expressed in terms of m_{n0}. The proof, however, contains a missing reduction: the passage from the full ideal m to its prime-to-n0 part is asserted without proof and supported only by an ambiguous citation. This is not an issue of disagreement with consensus or a technical style slip; it is a genuine unproved step in the central argument. The rest of the argument depends on this reduction to attach the cusp-form estimate to the original Hecke orbit. I therefore do not see a reason to move the reader's conditional verdict: the paper should be accepted only after Proposition 2.3 is either proved in the text or replaced by a precise reference with the required argument. The concrete computational check would settle whether the asserted reduction is true in general, and if it is true, would clarify which Brandt-matrix identity is needed.","tokens_in":115,"tokens_out":7415,"duration_ms":136136,"concrete_test":"Verify Proposition 2.3 directly: compute, for a concrete ramified prime p | n0, the Brandt matrix B(p) and check whether it is a permutation matrix. For example, take q = 3 and D ramified at t and ∞, compute B(t) from the explicit quaternion order, and confirm that B(t) has exactly one entry equal to 1 in each row and each column and zeros elsewhere. Then check the full identity B_ij(m)/σ_{n0}(m) = B_{kj}(m_{n0})/σ_{n0}(m_{n0}) for several m = p^v m_{n0}. If B(p) is not a permutation matrix, Proposition 2.3 is false; if it is, locate the exact statement in [WY11] or [Men12] that supplies the proof and fill in the missing argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4 controls |B_{kj}(m_{n0})/σ_{n0}(m_{n0}) − 1/(w_j deg e*)| via the Fourier coefficient c_{g_{kj}}(m_{n0}) and the Deligne–Ramanujan bound (3.4). The quantity needed for Theorem A is B_{ij}(m)/σ_{n0}(m) with m = m_{n0} ∏_{p|n0} p^{v_p(m)}. The only bridge is Proposition 2.3, which asserts that for each i there is a k with B_{ij}(m)/σ_{n0}(m) = B_{kj}(m_{n0})/σ_{n0}(m_{n0}) for all j. Since σ_{n0}(m) = σ_{n0}(m_{n0}), this is equivalent to saying that row i of B(m) equals row k of B(m_{n0}). Given the Hecke multiplicativity B(m) = B(∏ p^{v_p}) B(m_{n0}), the reduction holds exactly if each B(p) with p | n0 is a permutation matrix (so B(p^v) has one nonzero entry per row/column). This is not proved in the text; the proof refers to 'the same argument given in [Men12]' and to an unspecified 'Proposition II.1', with no location. If B(p) instead has multiple nonzero entries or entries larger than 1, the reduction fails and the final estimate applies only to the prime-to-n0 part, not to the Hecke orbit indexed by the full ideal m. This is load-bearing because without Proposition 2.3, the stated convergence of δ_{t_m e_i} is never established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem A: for a definite quaternion algebra over F_q(t) with discriminant n_0, the Hecke translates t_m e_i of a component e_i in the Picard group of the Shimura curve X_{n_0} become equidistributed with respect to the measure μ = δ_{e*} as deg m_{n_0} tends to infinity, where e* = Σ_i w_i^{-1} e_i. The proof follows the strategy of Menares: it expresses the Hecke action via Brandt matrices, compares the degree of t_m e_i with the divisor function σ_{n_0}, decomposes the theta series Θ_{ij} into a cuspidal part plus an Eisenstein part using the Wei–Yu isomorphism, and then applies Deligne–Ramanujan bounds to the Fourier coefficients of the cuspidal part. Theorem B is stated as a corollary for supersingular Drinfeld modules of rank 2, using the bijection between components and these modules and the mass formula.","tokens_in":6945,"tokens_out":9749,"duration_ms":90042,"significance":"If the proof is completed, the result is a natural function-field analogue and generalization of Menares' equidistribution theorem for Hecke points on the supersingular module, with a canonical target measure and no fitted parameters. The manuscript is concise and builds on substantial external results (Wei–Yu, Gekeler, Drinfeld), and the automorphic method used here is well suited to the problem. However, the paper as written has a load-bearing gap: Proposition 2.3, which is essential to pass from the estimate at the prime-to-n_0 part m_{n_0} back to the full ideal m, is not proved, and the cited reference is not specific enough to verify the claim. This prevents the main theorem from being established as stated.","major_comments":[{"comment":"Proposition 2.3 is the only bridge from the Fourier-coefficient estimate at m_{n_0} in Section 4 to the ratio B_{ij}(m)/σ_{n_0}(m) for the full ideal m = m_{n_0} ∏_{p|n_0} p^{v_p(m)}. The proof is not supplied: the text says 'This can be proven using the same argument given in [Men12] at the beginning of section 1.2' and refers to an unspecified 'Proposition II.1', without a precise location in the manuscript or in the reference list. The assertion is nontrivial: it says that for each i there is a k such that the normalized rows of B(m) and B(m_{n_0}) coincide, which requires control of the factors B(p) for p | n_0 that divide the level. Without this step, the estimate in Section 4 applies only to ideals coprime to n_0, and the convergence of the full Hecke orbit is not demonstrated. Please provide a complete proof of Proposition 2.3, or a precise statement-and-proof reference.","section":"Section 2, Proposition 2.3"},{"comment":"As typeset, the constant Fourier coefficient of Θ_{ij} is given by c_{Θ_{ij}}(r,0) = q^{-r} w_j in Section 3.3, but the constant term of E_{n_0} is given as q^{-r} Σ_i 1/w_i, and the computation in Section 4 uses c_{Θ_{ij}}(r,0) = q^{-r}/w_j to obtain c_{ij} = 1/(w_j deg(e*)). If the displayed q^{-r}w_j is not a typographical error, then the formula for E_{n_0} and the computation of c_{ij} are inconsistent. Since the identification of the Eisenstein component c_{ij} is essential for the main estimate, please correct the constant term and verify the normalization carefully.","section":"Sections 3.3 and 4, constant Fourier coefficient"}],"minor_comments":[{"comment":"The symbol σ_p appears in the main estimate where σ_{n_0} is meant; as written, the displayed chain mixes σ_p(m), σ_p(m_{n_0}), and σ_{n_0}(m_{n_0}). Please use one consistent notation.","section":"Section 4"},{"comment":"After applying Proposition 2.3, the convergence is first obtained for B_{kj}(m_{n_0})/σ_{n_0}(m_{n_0}); the final sentence should explicitly restate the conclusion for the original ratio B_{ij}(m)/σ_{n_0}(m) so that the role of Proposition 2.3 is transparent.","section":"Section 4, final paragraph"},{"comment":"The bound |δ_{t_m e_i}(f) - δ_{e*}(f)| ≤ max_j {f(e_j)} Σ_j |B_{ij}(m)/σ_{n_0}(m) - 1/(w_j deg e*)| is not valid for complex-valued f unless the maximum is replaced by max_j |f(e_j)|. Please correct this minor inequality.","section":"Section 2, inequality after Propositions 2.1 and 2.2"},{"comment":"The Hecke operators are first described for level Γ_0(p), but later they are used for Γ_0(n_0). This is presumably a typo for a general level, but the notation should be made uniform to avoid ambiguity.","section":"Section 3.2"},{"comment":"The reference to 'Proposition II.1' is not locatable in the manuscript or in the bibliography; if this is a result in [WY11] or [Men12], please give the exact theorem or proposition number and state the relevant statement.","section":"Section 2, Proposition 2.3"},{"comment":"The function σ_{n_0} is defined on elements of F_q[t] while Brandt matrices and Hecke operators are indexed by ideals; please clarify the translation by taking a monic generator of each ideal.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main issue is Proposition 2.3. I could not verify it from the cited reference as given, and the claim is essential because the Section 4 estimate only controls the prime-to-n_0 part. If the authors can supply a complete proof of Proposition 2.3 and fix the constant-term typo, the result appears to be sound and publishable. I would recommend insisting on those fixes before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper proves the function-field analogue of Menares's equidistribution theorem for Hecke orbits on the Picard group of definite Shimura curves, with a corollary for supersingular Drinfeld modules. The theorems are new, and the automorphic method is a good fit: Brandt matrices encode the Hecke movement, and Deligne-Ramanujan bounds for Drinfeld-type cusp forms give the decay. The transmission of the method from [Men12] is transparent and the cited inputs (WY11, the mass formula) are appropriate.\n\nThe soft spots are all in the exposition. Proposition 2.3 is load-bearing and its proof is not given, with a reference to an unlocated 'Proposition II.1'. The good news is that the reduction is true and easy: for p | n0, Proposition 2.2 gives the row sum σ_{n0}(p)=1, and the Brandt matrices are integer, so B(p) is a permutation matrix; the product of these over p|n0 is also a permutation, hence B(m) = B(m_{n0}) P and each row of B(m) is a row of B(m_{n0}). The authors should put this argument in the text.\n\nThere are also notational slips: the constant term of Θ_ij is printed as q^{-r} w_j in Section 3.3 but the computation in Section 4 needs q^{-r}/w_j (otherwise the Eisenstein series constant term doesn't add up). And in Section 4, σ_p appears where σ_{n0} is meant. These are typos, not mathematical errors.\n\nThe stress-test concern about Proposition 2.3 is therefore real for the text as written, but not for the mathematics: the reduction cannot fail because of the row-sum/integrality argument. Still, the missing proof is a genuine gap that a referee should require the authors to fill.\n\nOverall, this is a nice, careful paper with a correct main idea. It deserves a serious referee, and I would accept it provisionally. For a reading group, I'd say maybe: the core is clear and short, but the typos might be a mild annoyance.","headline":"Solid function-field analogue of Menares's theorem with a fillable gap: the key reduction is true but the authors must prove it instead of citing an unlocated 'Proposition II.1'.","tokens_in":7418,"tokens_out":17982,"would_cite":true,"duration_ms":154276,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G18","11G09","11F72","11R58"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that Hecke orbits on the Picard group of definite Shimura curves over the function field $\\mathbb{F}_q(t)$ equidistribute toward the measure $\\delta_{e^*}$, where $e^* = \\sum_i w_i^{-1} e_i$, and derives the…","keywords":["Hecke orbits","equidistribution","definite Shimura curves","Brandt matrices","Drinfeld modules","function fields","automorphic forms of Drinfeld type","supersingular Drinfeld modules"],"falsifier":"Compute the Brandt matrices for a small example, say $q=3$ and $n_0$ a prime of degree 1, for all monic $m$ with $\\deg m \\le 5$; for each $i,j$, check whether $B_{ij}(m)/\\sigma_{n_0}(m)$ tends to $1/(w_j\\deg e^*)$ as $\\deg m_{n_0}$ grows. A single ratio failing to approach that value would refute the equidistribution claim, and a counterexample to the claimed equality with $B_{kj}(m_{n_0})/\\sigma_{n_0}(m_{n_0})$ would show the proof's reduction does not hold in that case.","tokens_in":54,"feed_emoji":"📐","tokens_out":9327,"duration_ms":186126,"temperature":0.7,"pith_summary":"This paper proves that Hecke orbits on the Picard group of definite Shimura curves over the rational function field $\\mathbb{F}_q(t)$ are equidistributed. Fixing any component $e_i$, the orbit $\\{t_m e_i\\}$ under the Hecke correspondence, when ordered by the degree of the part of $m$ coprime to the discriminant $n_0$, becomes equidistributed with respect to the measure $\\delta_{e^*}$, where $e^*=\\sum_i w_i^{-1}e_i$. In the rank-2 supersingular case, this becomes the equidistribution of Hecke orbits $\\{T_m\\varphi\\}$ on the set of supersingular Drinfeld modules, with limiting masses proportional to $1/w_i$. The proof is automorphic: Brandt matrix coefficients are identified with Fourier coefficients of cuspidal automorphic forms of Drinfeld type, and Deligne-Ramanujan bounds make those coefficients tend to zero fast enough.","feed_headline":"Hecke orbits spread evenly on definite Shimura curves","feed_subtitle":"Function-field proof pins down the limiting spread of Hecke orbits, including supersingular Drinfeld modules.","key_machinery":"Three linked objects carry the argument. The Brandt matrix $B(m)=(B_{ij}(m))$ records the number of integral elements of a definite quaternion algebra with a given norm, and Proposition 2.1 translates the Hecke action into matrix multiplication, so $t_m e_i=\\sum_j B_{ij}(m)e_j$; the goal becomes showing that the normalized columns converge coefficientwise. Automorphic forms of Drinfeld type are functions on the oriented edges of the Bruhat-Tits tree of $\\mathrm{PGL}_2(k_\\infty)$ with a harmonicity condition, and the $\\theta$ series $\\Theta_{ij}$ have Fourier coefficients exactly $q^{-(\\deg m+2)}B_{ij}(m)$, connecting combinatorial counts to automorphic coefficients. The Deligne-Ramanujan bound for cuspidal automorphic forms of Drinfeld type, $|c_f(m)|\\ll q^{(\\varepsilon-1/2)\\deg m}$, supplies the decay that forces the ratios to their limits. A Hecke-equivariant decomposition theorem for the space generated by the $\\Theta_{ij}$ lets the paper isolate the cuspidal part, whose coefficients vanish in the limit.","core_discovery":"The central claim is Theorem A: for every index $i$ and every function $f$ on the finite set $B=\\{e_1,\\dots,e_n\\}$, the normalized sums $\\delta_{t_m e_i}(f)$ converge to $\\delta_{e^*}(f)$ as $\\deg m_{n_0}\\to\\infty$, where $m_{n_0}$ is the part of $m$ coprime to the discriminant of the quaternion algebra. The proof establishes the stronger quantitative estimate that the difference $|B_{ij}(m)/\\sigma_{n_0}(m)-1/(w_j\\deg e^*)|$ is $O(q^{(\\varepsilon-1/2)\\deg m_{n_0}})$ for every $\\varepsilon>0$, which forces the ratios of Brandt-matrix coefficients to their limiting values. When $n_0=p$ is a single prime, the same result gives Theorem B: the Hecke orbit $\\{T_m\\varphi\\}$ of any supersingular Drinfeld module of rank 2 becomes equidistributed on the full set $D_p^{ss}$ with respect to the measure $\\delta_\\Phi$, whose mass at $\\varphi_i$ is $1/w_i$ up to normalization. This is stated as a direct consequence of Theorem A through the bijection between divisor classes and supersingular Drinfeld modules.","pith_inferences":["Editorial extension: the same method may prove equidistribution for higher-rank Drinfeld modules or for other finite arithmetic quotients where Brandt-matrix analogues and Deligne-Ramanujan bounds are available.","Editorial extension: the quantitative bound suggests one can extract an explicit rate of convergence with the constant depending on $f$, $q$, and $n_0$; the paper does not compute such a constant.","Editorial extension: the pattern that the ramified part of the ideal is irrelevant to the limiting behavior may also appear in other function-field equidistribution problems, so the argument points toward a general principle that only the coprime part $m_{n_0}$ controls the spread.","Editorial extension: since the reduction in Proposition 2.3 is deferred to a reference, a direct proof of that reduction for general $m$ would remove the main deferred step; conversely, a small computational search for counterexamples to the reduction would test whether the present proof closes."],"forward_implications":["The finite Picard group $B$ receives a limiting measure $\\delta_{e^*}$ whose mass at $e_i$ is inversely proportional to the order of the unit group of the corresponding maximal order; the equidistribution statement therefore describes exactly how often each component is visited.","For the supersingular Drinfeld module interpretation, the limiting measure is completely explicit: the mass at each supersingular module $\\varphi_i$ is $1/w_i$, and the total mass gives the class-number normalization $(q^{\\deg p}-1)/(q^2-1)$.","The proof yields an exponential rate of convergence $q^{(\\varepsilon-1/2)\\deg m_{n_0}}$ for all $\\varepsilon>0$, so the deviation from the limit is bounded by a power of $q^{-\\deg m_{n_0}}$ with exponent close to $1/2$.","The automorphic approach is robust enough that the same argument applies to all definite quaternion algebras over $\\mathbb{F}_q(t)$, not only to those of prime discriminant; the paper writes the proof for arbitrary $n_0$."],"supporting_citations":[{"why":"Constructs the Shimura curves, introduces Brandt matrices, and proves the Hecke-equivariant decomposition of the theta series used to identify Fourier coefficients with Brandt entries.","marker":"[WY11]"},{"why":"Gives the proof strategy for Hecke point equidistribution and is the source to which Proposition 2.3 defers its reduction argument.","marker":"[Men12]"},{"why":"Establishes the Ramanujan bound on Frobenius eigenvalues, from which the coefficient bound for cuspidal automorphic forms of Drinfeld type is derived.","marker":"[Dri74]"},{"why":"Provides the Fourier expansion of automorphic forms of Drinfeld type and the linear recurrences linking Hecke eigenvalues to Fourier coefficients.","marker":"[Gek95]"},{"why":"Supplies the basis of newforms and the Deligne-Ramanujan bounds for their Hecke eigenvalues.","marker":"[GR96]"},{"why":"Gives the bijection between divisor classes on the Shimura curve and supersingular Drinfeld modules used to state Theorem B.","marker":"[Pap05]"},{"why":"Provides the mass formula for supersingular Drinfeld modules that fixes the normalization of the limiting measure in Theorem B.","marker":"[Gek83]"},{"why":"Defines the Hecke correspondence $t_m$ on the Picard group in the function-field setting.","marker":"[CWY17]"}],"fun_headline_variants":["Hecke orbits equidistribute on Picard groups","Equidistribution for Hecke orbits on definite Shimura curves","Hecke orbits spread uniformly on Picard group","Function-field proof: Hecke orbits equidistribute","Definite Shimura curves: Hecke orbits spread evenly"],"cache_read_input_tokens":9472,"weakest_assumption_plain":"The proof relies on a reduction, stated as Proposition 2.3 without a full proof, which says that every ratio of counts for an ideal $m$ can be replaced by the same kind of ratio for the part of $m$ coprime to the ramification locus; if that replacement has any exception, the convergence argument does not close.","fun_headline_variants_meta":{"raw":{"variants":["Hecke orbits equidistribute on Picard groups","Equidistribution for Hecke orbits on definite Shimura curves","Hecke orbits spread uniformly on Picard group","Function-field proof: Hecke orbits equidistribute","Definite Shimura curves: Hecke orbits spread evenly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000221,"raw_usage":{"total_tokens":1417,"prompt_tokens":882,"completion_tokens":535,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":457}},"tokens_in":498,"tokens_out":535,"duration_ms":5527,"temperature":1.0,"reasoning_tokens":457,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:53:08.634502+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Brandt matrices for a small example, say $q=3$ and $n_0$ a prime of degree 1, for all monic $m$ with $\\deg m \\le 5$; for each $i,j$, check whether $B_{ij}(m)/\\sigma_{n_0}(m)$ tends to $1/(w_j\\deg e^*)$ as $\\deg m_{n_0}$ grows. A single ratio failing to approach that value would refute the equidistribution claim, and a counterexample to the claimed equality with $B_{kj}(m_{n_0})/\\sigma_{n_0}(m_{n_0})$ would show the proof's reduction does not hold in that case.","supporting_citations":[],"review_version":1}