{"id":"eaaafd9f-5ef8-4f28-b72b-a7c5f5d58ef2","arxiv_id":"1908.04248","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Concomitants of ternary quartics parametrize all vector-valued Siegel modular forms of degree 3, with boundary vanishing orders matched to double-conic vanishing orders.","lead":"This paper shows that all vector-valued Siegel and Teichmüller modular forms of genus three can be built from the representation theory of plane quartic curves. It also relates Teichmüller cusp forms to the cohomology of local systems on moduli of curves, with many explicit cases in genus 3.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 11.6 depends on Proposition 11.1, whose proof invokes an uncited blow-up model of M3 and asserts a factor-2 relation between the exceptional divisor and the hyperelliptic locus; if this geometric input fails, the central isomorphism does not hold.","rationale":"The reader's weakest-assumption analysis identifies Proposition 11.1 and its unsupported use of a blow-up model of M3; my reading agrees. This is genuinely load-bearing because Theorem 11.6 is the paper's first main result, and its proof uses Proposition 11.1 both to place the image in S^n and to prove surjectivity by writing β(F)/Ξ^n as a regular concomitant. The factor 2 in the order formula is not a harmless normalization: it determines n = d − 2m, and a wrong factor would change every dimension statement and every Hecke-eigenvalue computation in Sections 12 and 13. At the same time, I do not find an internal inconsistency in the rest of the argument: the Fourier-expansion computations for χ4,0,8, the substitution map γ', and the examples in Section 12 are mutually consistent, and the Hecke checks against [5] provide real evidence. The paper also has independent support in the form of reproducible-looking coefficient computations and agreement with known one-dimensional spaces such as S4,0,8 and S1,8,5. The concern is therefore that a central but unproved geometric assertion may fail, not that the paper is demonstrably wrong. Since the reader already conditioned the verdict on this point, I see no reason to move the verdict. A useful next step is the concrete check above: either the recalled model appears in the literature with the stated properties, or Proposition 11.1 needs a full proof.","tokens_in":33544,"tokens_out":12778,"duration_ms":152374,"concrete_test":"Verify the recalled birational model and the factor-2 claim directly: consult the standard GIT construction for plane quartics (e.g., Mumford-Fogarty or Fedorchuk-Smyth [18]) to check whether M3 is obtained from P14 by a single blow-up of DC followed by deleting the discriminant and quotienting by PGL(3,C), and compute the local ramification degree of the exceptional divisor over the hyperelliptic locus. Independently, take the family f_t = t f + g^2 with g a general conic and compute the order in t of the universal quartic concomitant f (d = 1, v = 0) and of the discriminant Ξ (d = 27, v = 14); the formula 2v − d must give −1 and +1 respectively, matching the known poles of χ4,0,−1 and χ9. If the geometric model requires additional blow-ups or the factor is not 2, recompute Proposition 11.1 with the corrected local coordinates.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central isomorphism of Theorem 11.6 is powered by Proposition 11.1, which states that a concomitant of degree d and order v along the double-conic locus DC yields a Teichmüller modular form of order 2v − d along the hyperelliptic locus. The proof is not self-contained: it says the result is analogous to [11, Theorem 1] and then 'recall[s] that the coarse moduli space M3 may be constructed by blowing up the locus of double conics in the projective space of ternary quartics, deleting the proper transform of the discriminant hypersurface, and taking the quotient by PGL(3,C)'. No reference or argument is supplied for this model, and the paper does not show why the exceptional divisor contributes a factor of 2 to orders of vanishing when passing from the quotient of the blow-up to the stack M3. The factor 2 is exactly what turns the double-conic order v into the hyperelliptic order 2v − d, and hence it controls the exponent n = d − 2m in Theorem 11.6. If the correct resolution of the period map requires more than one blow-up, or if the exceptional divisor maps to H3 with a different ramification degree, the formula for n and the entire dictionary between concomitants and Siegel modular forms would shift. This is the weakest load-bearing link in the paper: the later examples are consistent with it, but they do not replace the missing geometric verification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a dictionary between the representation theory of ternary quartics and vector-valued Siegel and Teichmüller modular forms of genus three. The central object is the map that substitutes the Fourier expansion of the vector-valued cusp form χ_{4,0,8} into a concomitant of ternary quartics. Theorem 11.6 asserts an isomorphism between the space C_{d,ρ}(-mDC) of concomitants of degree d, type ρ, vanishing to order at least m along the double-conic locus and the space S^n_{ρ1-ρ2,ρ2-ρ3,ρ3+9n} of Siegel modular forms vanishing to order at least n along the boundary, with n=d-2m. This theorem is powered by Proposition 11.1, which relates orders of vanishing along the double-conic locus to orders along the hyperelliptic locus. The second main result, Theorem 13.1, identifies the top Hodge-filtered pieces of the middle cohomology of symplectic local systems on M_g with spaces of Teichmüller cusp forms satisfying prescribed vanishing conditions on the boundary. The paper also contains many explicit Fourier expansions, Hecke eigenvalue checks against the authors' earlier database [5], and an appendix proving that Teichmüller modular forms extend to the Deligne-Mumford compactification for g≥3.","tokens_in":33853,"tokens_out":4967,"duration_ms":55123,"significance":"If Theorems 11.6 and 13.1 hold, the paper gives a constructive and essentially complete description of vector-valued Siegel and Teichmüller modular forms of degree three, and it establishes a new bridge between Teichmüller cusp forms and the cohomology of local systems on M_g for arbitrary g. The main strengths are the explicit Fourier-Jacobi expansion obtained from the Schottky form, the careful reduction of Teichmüller forms to Siegel forms via χ_9, and the precise extension statement proved in Section 14. The Hecke eigenvalue checks and the agreement with [5] provide concrete numerical support for the constructions. At the same time, the central geometric input in Proposition 11.1 is not proved and is invoked by analogy; since this input controls the exponent n=d-2m in Theorem 11.6, the central claim is not yet fully established. The paper would be a strong contribution after this gap is filled and after the deferred computational details are made available.","major_comments":[{"comment":"The proof of the order-of-vanishing formula is not self-contained. The paragraph beginning \"Now recall that the coarse moduli space M_3 may be constructed by blowing up the locus of double conics...\" asserts, without proof or citation, a specific blow-up model of M_3 and a factor-2 relation between the exceptional divisor and the hyperelliptic locus. This factor is exactly what converts the double-conic order v into the hyperelliptic order 2v-d, and therefore it determines the exponent n=d-2m in Theorem 11.6. If the correct resolution requires more than one blow-up, or if the ramification degree along the hyperelliptic locus differs from 2, the formula for n and the isomorphism of Theorem 11.6 would shift. A rigorous proof or a precise reference for this geometric model is required before the main theorem can be considered established.","section":"§11, Proposition 11.1"},{"comment":"The proof of Theorem 13.1 relies on the assertion that \"the spectral sequence associated to the Hodge filtration degenerates at E1\" for the logarithmic de Rham complex that computes the cohomology of the local system V'_μ on M_g and Mc_g. This degeneration is a nontrivial input for a general symplectic local system on M_g, and no reference or argument is supplied in the proof. Since Theorem 13.1 is one of the two main results of the paper, the missing justification should be supplied, for example by citing the relevant Hodge-theoretic degeneration theorem for logarithmic de Rham complexes or by giving a direct argument in this setting.","section":"§13, Theorem 13.1"},{"comment":"The paper states that \"we have explicitly computed all (spaces of) concomitants of ternary quartics of degree at most 6\" and \"the computations are quite involved and we will discuss them in a future paper.\" These computations are used to support the claimed identifications of motives and the conjectural Euler characteristics in the following paragraphs. Because the supporting data are not included, those particular claims cannot be independently checked. This does not affect the main theorems, but it limits the verifiability of the evidence presented for the cohomological conjectures.","section":"§13, after the list of d=4 cases"}],"minor_comments":[{"comment":"The Hecke eigenvalue checks for λ_3 and λ_5 are asserted but the corresponding coefficients and formulas are not shown; providing them would make the numerical verification more transparent.","section":"§5"},{"comment":"The statement that an odd Teichmüller modular form \"vanishes with multiplicity at least three along δ1\" is used later but the reason (divisibility by χ_9, whose divisor is h+δ_0+3δ_1) could be stated explicitly at that point for clarity.","section":"§7, Corollary 7.5"},{"comment":"The line bundles O(13λ-δ), O(13λ+δ_0-2δ), and O(13λ-2δ) on \\overline{M}_g are used without specifying the convention for δ and its components; a sentence defining δ, δ_0, and δ_1 would prevent ambiguity.","section":"§13, equations (14)-(16)"},{"comment":"The deduction that the catalecticant form has order at least 16 along A_{2,1} because the first cusp form on Γ_1 vanishing with order 6 at infinity is Δ^6 is terse; the relation between weights and vanishing orders along the boundary factor should be spelled out.","section":"§12.4"},{"comment":"The title and abstract contain apparent typographical artifacts such as \"V ALUED\"; these should be corrected in the final version.","section":"Title and abstract"}],"recommendation":"major_revision","confidential_remarks":"The numerical checks against [4] and [5] are partly checks against the authors' own earlier computations and conjectures. This is not inappropriate, but it means the agreement is less independent than a comparison with an external database would be; the editor may wish to weigh this when assessing the strength of the evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nTwo things to know before you read this paper. First, it is a serious piece of work: it gives the first complete constructive description of all vector-valued Siegel and Teichmüller modular forms of genus three, as a module over the scalar ring, via concomitants of ternary quartics. Second, the weakest link is Proposition 11.1, a key input to the main structural theorem, whose proof invokes a blow-up model of M_3 without citation or argument.\n\nWhat is genuinely new: Theorem 11.6 is a clean statement — spaces of concomitants with order conditions along the double-conic locus are isomorphic to spaces of Siegel modular forms with order conditions at infinity. The substitution map c -> c∘χ_{4,0,8} makes the construction completely explicit. Theorem 13.1, for arbitrary genus, identifies top Hodge pieces of middle cohomology of symplectic local systems on M_g with Teichmüller cusp forms with prescribed vanishing along the boundary; that is new, and the log de Rham proof is straightforward and mostly self-contained. The paper also ships a lot of explicit data: Fourier expansions, Hecke eigenvalue checks against the authors' own database [5], and the motive identifications for small weights are consistent with the point-counting results of [4]. That reproducible computational core is real evidence.\n\nThe soft spot is exactly the one the stress-test flags. Proposition 11.1 asserts that the order along the hyperelliptic locus equals 2v - d, with the factor 2 attributed to “the difference between the stack M_3 and M_3” via a blow-up of the double-conic locus. The proof is by analogy with the binary sextic case [11, Theorem 1]; the blow-up model is simply “recalled”; and the ramification factor is not justified. The later examples are consistent with the formula, but they do not replace the missing geometric verification. If the factor were different, the dictionary n = d - 2m in Theorem 11.6 would shift. This is not a fatal objection — the surrounding evidence is strong — but it is an honest “needs verification” flag, and it is the one place I would ask the authors to write out the geometry before trusting the statement in full.\n\nWho gets value from this paper: anyone working on Siegel modular forms of low genus or on the cohomology of local systems on M_g. Sections 7–11 give a useful map; Section 14, on extension of Teichmüller forms to the Deligne-Mumford compactification, is a neat standalone appendix. The paper deserves a serious referee — the structural claims are important and mostly proven, and the one soft spot is localizable and fixable. I would engage with it.\n\nBest,","headline":"A genuinely new structural result on degree-3 Siegel and Teichmüller modular forms, with a key order-of-vanishing bridge (Prop. 11.1) that leans on an uncited blow-up model and needs a fill-in before the main theorem is fully load-bearing.","tokens_in":34364,"tokens_out":2715,"would_cite":true,"duration_ms":29056,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F46","14H10","14H45","14J15","14K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes an explicit isomorphism that turns covariants of plane quartics into all vector-valued Siegel and Teichmüller modular forms of genus 3, with vanishing orders translated across a dictionary.","keywords":["ternary quartics","concomitants","vector-valued Siegel modular forms","Teichmüller modular forms","genus three","double conics","Schottky form","Hodge bundle"],"falsifier":"Test Proposition 11.1 on the classical degree-2 covariant $\\sigma$: compute its vanishing order $v$ along the double-conic locus by degenerating a quartic to $tf+g^2$, and independently compute the order of the resulting Siegel modular form along the hyperelliptic locus from its Fourier expansion near that locus. The paper predicts $\\chi_{0,4,16}$ is not divisible by $\\chi_{18}$; a finding that it vanishes on the hyperelliptic locus would refute the equality $\\mathrm{ord}_{H_3}(\\gamma(c)) = 2v-d$ and with it the main isomorphism.","tokens_in":33373,"feed_emoji":"📐","tokens_out":10588,"duration_ms":99935,"temperature":0.7,"pith_summary":"The paper proves that every vector-valued Siegel or Teichmüller modular form of genus 3 can be produced by a classical invariant-theoretic machine: take a concomitant of ternary quartics, an equivariant polynomial expression in the coefficients of a plane quartic with values in a representation of $\\mathrm{GL}(3)$, and substitute into it the fifteen Fourier coefficients of one fixed cusp form $\\chi_{4,0,8}$. The key interchange is an order-of-vanishing equality: a concomitant vanishing to order $v$ on the locus of double conics yields a modular form vanishing to order $2v-d$ on the hyperelliptic locus. This yields an explicit isomorphism between spaces of concomitants with fixed vanishing along double conics and spaces of vector-valued Siegel modular forms with fixed vanishing at the boundary. A second result identifies the top Hodge pieces of the middle cohomology of symplectic local systems on moduli spaces of curves with spaces of Teichmüller cusp forms, extending the classical description known for abelian varieties. If correct, the construction makes the modular forms explicit enough to compute Fourier expansions and Hecke eigenvalues by polynomial substitution.","feed_headline":"Ternary quartics encode all genus-3 modular forms","feed_subtitle":"A single cusp form and one substitution rule turn classical concomitants into explicit Siegel and Teichmüller forms.","key_machinery":"The central object is the concomitant of ternary quartics: an equivariant polynomial section of a symmetric power of the space of ternary quartics, valued in an irreducible representation of $\\mathrm{GL}(3)$. The carrying identity is the order formula of Proposition 11.1, which translates vanishing along the double-conic locus into vanishing along the hyperelliptic locus. The substitution map $\\gamma$ sends a concomitant $c$ to $c(\\chi_{4,0,-1})$, where $\\chi_{4,0,-1}=\\chi_{4,0,8}/\\chi_9$ is the meromorphic image of the universal quartic; replacing each quartic coefficient by the normalized Fourier coefficient $\\alpha_I$ of $\\chi_{4,0,8}$ gives the holomorphic form $\\gamma'(c)=c(\\chi_{4,0,8})$. The Fourier expansion of $\\chi_{4,0,8}$ itself is obtained by developing the Schottky form $J_8$ along $\\mathbb{H}_3\\times\\mathbb{H}_1$, whose first Fourier–Jacobi coefficient is an explicit product of $\\theta$ constants, and it is identified geometrically with the quartic cutting out the canonical model of a nonhyperelliptic genus-3 curve.","core_discovery":"The central claim is an isomorphism of vector spaces\n$$C_{d,\\rho}(-m\\,\\mathrm{DC}) \\xrightarrow{\\ \\sim\\ } $S^{{\\,n}}$_{\\rho_1-\\rho_2,\\ \\rho_2-\\rho_3,\\ \\rho_3+9n}, \\qquad n = d - 2m,$$\nsending a concomitant $c$ to $c(\\chi_{4,0,-1})\\chi_9^n$, or equivalently substituting the Fourier coefficients of $\\chi_{4,0,8}$ and clearing poles. Here $C_{d,\\rho}(-m\\,\\mathrm{DC})$ is the space of concomitants of ternary quartics of degree $d$ and type $\\rho$ vanishing to order at least $m$ along the double-conic locus, and $S^n_{i,j,k}$ is the space of vector-valued Siegel modular forms of weight $(i,j,k)$ vanishing to order at least $n$ along the boundary divisor. The proof runs through a dictionary: a concomitant with vanishing order $v$ along double conics produces a meromorphic modular form with order $2v-d$ along the hyperelliptic locus. Because every holomorphic form is obtained this way, the paper concludes that all vector-valued Siegel and Teichmüller modular forms of degree 3 are explicit polynomials in the Fourier expansion of the single form $\\chi_{4,0,8}$.","pith_inferences":["Inference: If the isomorphism is as complete as stated, dimension tables for all genus-3 vector-valued Siegel cusp forms could in principle be produced by finite linear algebra on ternary quartic representations, bypassing trace-formula calculations; comparing the two for an uncomputed weight would be a direct test.","Inference: The same substitution picture suggests a general recipe for low-genus moduli spaces realized as quotients of a space of forms: modular forms should be the associated graded ring of covariants under a valuation measuring vanishing along the exceptional divisor, with the double-conic order playing that role here.","Inference: Since $\\chi_{4,0,8}$ is also the theta section whose lowest-order term cuts out the canonical quartic, the substitution $c \\mapsto c(\\chi_{4,0,8})$ may be read geometrically as evaluating the concomitant on the universal canonical curve, which could connect Fourier expansions of genus-3 forms to theta functions and to the hyperelliptic locus more directly than the paper spells out."],"forward_implications":["Every holomorphic vector-valued Siegel cusp form of degree 3 admits a constructive expression as a polynomial in the Fourier coefficients of $\\chi_{4,0,8}$; the paper carries this out in many cases and checks the resulting Hecke eigenvalues.","Dimension and vanishing data for $S^n_{i,j,k}$ become questions about the representation theory of ternary quartics; Corollary 11.7 gives a concrete vanishing range $i+2j+4k < 36n \\Rightarrow S^n_{i,j,k}=0$.","The square of $\\chi_{18}$ is the unique generator of the space of scalar cusp forms of weight $18k$ vanishing to order $2k$ at the boundary, a broader form of the classical uniqueness of $\\chi_{18}$.","For genus 3, odd Teichmüller modular forms are exactly $\\chi_9$ times pullbacks of Siegel forms, so the full ring of Teichmüller forms is known once the Siegel side is known.","The top Hodge pieces of the middle cohomology of symplectic local systems on $M_g$ are isomorphic to spaces of Teichmüller cusp forms with prescribed boundary vanishing, and in genus 3 the paper matches these spaces with point counts over finite fields."],"supporting_citations":[{"why":"sets up the concomitant formalism for ternary quartics and gives the representation decompositions used throughout","marker":"[9]"},{"why":"the proof of Proposition 11.1 is modeled on its Theorem 1, supplying the order-of-vanishing argument for binary sextics","marker":"[11]"},{"why":"provides the Fourier–Jacobi development of the Schottky form and the Fourier expansion of $\\chi_{4,0,8}$ that the substitution map uses","marker":"[12]"},{"why":"gives the theta construction whose lowest-order term is the canonical quartic, identifying the universal quartic with $\\chi_{4,0,8}$","marker":"[19]"},{"why":"establishes the Teichmüller cusp form $\\chi_9$ whose powers clear the poles in the substitution map","marker":"[26]"},{"why":"develops Teichmüller modular forms of degree 3, including the generation result used to reduce Teichmüller forms to Siegel forms","marker":"[27]"},{"why":"supplies the degeneration and cohomological framework that the second main theorem adapts from abelian varieties to curves","marker":"[17]"},{"why":"provides cohomological computations and predicted Hecke eigenvalues used to verify the constructed modular forms","marker":"[4]"},{"why":"describes the ring of scalar Siegel modular forms of degree 3, used for dimensions and generators in examples","marker":"[43]"},{"why":"gives dimension formulas for level-one vector-valued Siegel forms used to identify spaces such as $S_{4,0,8}$ and to check dimensions","marker":"[42]"}],"fun_headline_variants":["One substitution rule yields every genus-3 modular form","All genus-3 Siegel and Teichmüller forms from one cusp form","Double-conic vanishing orders determine all genus-3 forms","Explicit construction: all genus-3 forms from one quartic","One quartic, all genus-3 modular forms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole dictionary assumes that the space of all genus-3 curves can be built by blowing up the locus of double conics in the space of plane quartics, removing the singular quartics, and quotienting by coordinate changes; no proof of that model is given, and if it fails, the key order-of-vanishing formula has no support and the main theorem falls.","fun_headline_variants_meta":{"raw":{"variants":["One substitution rule yields every genus-3 modular form","All genus-3 Siegel and Teichmüller forms from one cusp form","Double-conic vanishing orders determine all genus-3 forms","Explicit construction: all genus-3 forms from one quartic","One quartic, all genus-3 modular forms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001383,"raw_usage":{"total_tokens":5599,"prompt_tokens":941,"completion_tokens":4658,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":4572}},"tokens_in":557,"tokens_out":4658,"duration_ms":34875,"temperature":1.0,"reasoning_tokens":4572,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:46:12.822321+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test Proposition 11.1 on the classical degree-2 covariant $\\sigma$: compute its vanishing order $v$ along the double-conic locus by degenerating a quartic to $tf+g^2$, and independently compute the order of the resulting Siegel modular form along the hyperelliptic locus from its Fourier expansion near that locus. The paper predicts $\\chi_{0,4,16}$ is not divisible by $\\chi_{18}$; a finding that it vanishes on the hyperelliptic locus would refute the equality $\\mathrm{ord}_{H_3}(\\gamma(c)) = 2v-d$ and with it the main isomorphism.","supporting_citations":[{"cited_title":"Chipalkatti: The Waring loci of ternary quartics","cited_arxiv_id":null,"evidence_quote":"sets up the concomitant formalism for ternary quartics and gives the representation decompositions used throughout"},{"cited_title":"Cl´ ery, C","cited_arxiv_id":null,"evidence_quote":"the proof of Proposition 11.1 is modeled on its Theorem 1, supplying the order-of-vanishing argument for binary sextics"},{"cited_title":"Cl´ ery, G","cited_arxiv_id":null,"evidence_quote":"provides the Fourier–Jacobi development of the Schottky form and the Fourier expansion of $\\chi_{4,0,8}$ that the substitution map uses"},{"cited_title":"Frobenius: ¨Uber die Jacobischen Functionen dreier Variabelen","cited_arxiv_id":null,"evidence_quote":"gives the theta construction whose lowest-order term is the canonical quartic, identifying the universal quartic with $\\chi_{4,0,8}$"},{"cited_title":"Ichikawa: On Teichm¨ uller modular forms.Math","cited_arxiv_id":null,"evidence_quote":"establishes the Teichmüller cusp form $\\chi_9$ whose powers clear the poles in the substitution map"},{"cited_title":"Ichikawa: Teichm¨ uller modular forms of degree 3","cited_arxiv_id":null,"evidence_quote":"develops Teichmüller modular forms of degree 3, including the generation result used to reduce Teichmüller forms to Siegel forms"},{"cited_title":"Faltings, C.-L","cited_arxiv_id":null,"evidence_quote":"supplies the degeneration and cohomological framework that the second main theorem adapts from abelian varieties to curves"},{"cited_title":"Bergstr¨ om, C","cited_arxiv_id":null,"evidence_quote":"provides cohomological computations and predicted Hecke eigenvalues used to verify the constructed modular forms"},{"cited_title":"Ta ¨ ıbi:Dimensions of spaces of level one automorphic forms for split classic al groups using the trace formula","cited_arxiv_id":null,"evidence_quote":"describes the ring of scalar Siegel modular forms of degree 3, used for dimensions and generators in examples"},{"cited_title":"Shioda: On the graded ring of invariants of binary octavics","cited_arxiv_id":null,"evidence_quote":"gives dimension formulas for level-one vector-valued Siegel forms used to identify spaces such as $S_{4,0,8}$ and to check dimensions"}],"review_version":1}