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Elliptic matroids and modular curves

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Pith's one-line read For every n ≥ 10, the open modular curve X_1(n)^∘ and the realization space of the elliptic matroid T_n are the same scheme over Z[1/n], making the matroid's realizations exactly the torsion configurations on smooth or nodal plane cubics.

desk verdict A solid and genuinely new scheme-theoretic extension of the Borisov–Roulleau correspondence; the central argument holds up, with only a minor completeness gap in the small-n section. read the letter →

arxiv 2608.05299 v1 pith:6NYEXJIX submitted 2026-08-05 math.AG math.CO

Matthew Baker This is my paper · ORCID
classification math.AGmath.CO MSC 05B3514H5214G3514N05
keywords ellipticmatroidsmodularcurvesmatroidrealizationspaceszero-sumgridpropagationplanecubicgrouplawArtiniandeformationsscheme-theoreticisomorphismrationaltorsionpoints
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

This paper proves that two objects studied in different branches of mathematics coincide for every $n \geq 10$: the open modular curve $X_1(n)^\circ$, whose points parameterize elliptic curves (or irreducible nodal generalized elliptic curves) equipped with a point of exact order $n$, and the realization space $R_n$ of the elliptic matroid $T_n$, the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose non-bases are the three-element subsets with sum zero. For every field $k$ with $\mathrm{char}(k) \nmid n$, a $k$-point of the modular curve maps bijectively to a rescaling class of $k$-realizations of $T_n$, and this correspondence is upgraded to an isomorphism of schemes over $\mathbb{Z}[1/n]$. The proof is purely algebraic and incidence-theoretic, using only projective geometry, intersection counting, and the group law on plane cubics, replacing an earlier complex-analytic comparison. A corollary reformulates the classical statement that no elliptic curve over $\mathbb{Q}$ has a rational point of prime order $p \geq 11$ as the non-representability of the elliptic matroid $T_p$ over $\mathbb{Q}$.

What carries the argument

The load-bearing construction is the zero-sum grid: a $3\times 3$ array of labels in $\mathbb{Z}/n\mathbb{Z}$ whose rows and columns each sum to zero and whose nine entries are pairwise distinct. The row lines and column lines through the corresponding marked points form two reducible cubics; their scheme-theoretic intersection is exactly the nine marked points, and the classical theorem on cubics through eight of nine intersection points forces any cubic through eight of them to pass through the ninth. A seed pencil through nine points, with a tenth point selecting a unique member, is then propagated across $\mathbb{Z}/n\mathbb{Z}$ by successively applying this completion to carefully chosen grids. The group law is recovered by encoding the collinearity relations as equations in the generalized Jacobian: the line-section class $\lambda = [H - 3P_0]$ is forced to vanish and each marked point becomes $iP_1$, with $P_1$ of exact order $n$. Scheme-theoretically, the same grid argument is run over local Artinian rings with relative lines and sections, using flatness and local algebra to promote field-level transversality, and an Artinian-point criterion upgrades the resulting bijection on all Artinian $A$-valued points to the global isomorphism.

What would settle it

Enumerate both sides of $\beta_k$ for a small explicit case, say $n=10$ and a finite field $k$ of characteristic not dividing $10$, and compare the number of $k$-points: a mismatch of cardinalities would disprove the field-valued bijection. Alternatively, take a normalized realization of $T_{10}$ over the dual numbers $k[\varepsilon]/(\varepsilon^2)$ and check whether its first-order deformation is realized by a deformation of the corresponding marked cubic; an Artinian deformation not coming from the modular side would falsify the scheme-theoretic isomorphism.

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Extended reading notes

Core claim

At the center of the paper is a two-way dictionary between torsion configurations and matroid realizations. Starting from a marked plane cubic $(E,O,P)$ with $P$ of exact order $n$, the points $iP$ for $i \in \mathbb{Z}/n\mathbb{Z}$ form a configuration in which three distinct points are collinear exactly when their labels sum to zero in $\mathbb{Z}/n\mathbb{Z}$; this is a realization of $T_n$. The paper proves the reverse direction: any realization of $T_n$ over a field $k$ of characteristic not dividing $n$ determines a unique cubic through all its points, that cubic is irreducible and the marked points are smooth on it, the smooth locus carries a group law with $P_0$ as identity, and the configuration is recovered as $P_i = iP_1$ with $\mathcal{O}_C(1) \cong \mathcal{O}_C(3P_0)$. The reconstruction works uniformly on local Artinian rings, which by an Artinian-point criterion yields the scheme-theoretic isomorphism $\beta: X_1(n)^\circ \to R_n$ over $\mathbb{Z}[1/n]$.

Load-bearing premise

The argument presupposes the standard theory that the $\Gamma_1(n)$ moduli problem is representable by the smooth scheme $X_1(n)$ over $\mathbb{Z}[1/n]$ and that on its open locus $X_1(n)^\circ$ the universal generalized elliptic curve is embedded in the projective plane by the complete linear system $|3O|$; if either the representability or the embedding failed, the morphism $\beta$ and the reconstruction would have no starting point.

Editorial extensions

If this is right

  • For every field $k$ with $\mathrm{char}(k) \nmid n$ and every $n \geq 10$, the collinearity pattern of $T_n$ characterizes exactly the torsion-point configurations of smooth or nodal plane cubics of level $n$.
  • The modular curve $X_1(n)^\circ$ acquires a natural affine model over $\mathbb{Z}[1/n]$ as the multidegree-zero coordinate ring of the matroid realization scheme, so arithmetic questions about this modular curve can be translated into matroid data.
  • For primes $p \geq 11$, the statement that the elliptic matroid $T_p$ is not representable over $\mathbb{Q}$ is equivalent to the classical theorem that no elliptic curve over $\mathbb{Q}$ has a rational point of order $p$.
  • The known complex-analytic comparison over $\mathbb{C}$ follows as a special case, without modular forms or computer-assisted computation.
  • The realization scheme has a canonical band-scheme model over $\mathbb{F}_1^\pm$ whose tropical points are the reduced Dressian of $T_n$, i.e., the valuated matroids with underlying matroid $T_n$ up to rescaling.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural testable extension would be to relax the matroid or change the modular model at primes $p$ dividing $n$: the paper shows the simple matroid realization must break there, since level sections need not give distinct points, but the isomorphism might survive in a modified form.
  • The reconstruction of a group law from purely collinearity data suggests a general rigidity principle: sufficiently large matroid configurations with prescribed collinearities can force an ambient algebraic-group structure, with possible analogues for other linear systems or higher-rank moduli problems.
  • Because the proof is purely incidence-theoretic, analogous zero-sum hypergraphs on other abelian groups could yield new modular-curve identifications or new realization-space models for related modular curves.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 8 minor

Summary. The paper defines, for n≥4, the rank-3 elliptic matroid T_n on Z/nZ whose non-bases are the three-element subsets of distinct elements summing to zero. It then proves two main results. Theorem 1.1 establishes a natural bijection β_k: X_1(n)^∘(k) → R_{T_n}(k) for every n≥10 and every field k with char(k)∤n, where X_1(n)^∘ is the open part of the modular curve whose generalized elliptic curves have irreducible geometric fibers and R_{T_n}(k) is the set of rescaling classes of k-realizations of T_n. Theorem 1.2 upgrades this field-valued bijection to an isomorphism of schemes β: X_1(n)^∘ → R_n over Z[1/n]. The proof of Theorem 1.1 reconstructs, from an arbitrary realization, a unique irreducible plane cubic through all marked points using a seed pencil, Chasles propagation through zero-sum grids, and a group-law recovery argument; the cases n=10 and n=11 are handled separately. Part 2 extends the reconstruction over local Artinian rings and applies an Artinian-point criterion for isomorphisms. An appendix builds a canonical band-scheme realization space over F_1^±. As a corollary, the non-representability of T_p over Q for primes p≥11 is shown to be equivalent to the prime-order case of Mazur's theorem.

Significance. If correct, this is a substantial generalization of the Borisov–Roulleau theorem: it replaces complex-analytic methods by an elementary algebraic and incidence-theoretic argument, extends the correspondence to all fields of characteristic prime to n, and gives a natural Z[1/n]-model of X_1(n)^∘ as a matroid realization space. The proof is unusually self-contained in Part 1, with the n=10 and n=11 cases worked out in detail, and the scheme-theoretic Part 2 is logically coherent. I specifically checked the points that a skeptic might worry about. The Artinian-point criterion is stated and proved in Section 14.1, the relative seed-pencil and Chasles arguments in Section 15 are supplied in detail rather than assumed, and the application of the criterion in Corollary 15.8 is legitimate because every local Artinian Z[1/n]-algebra has residue field of characteristic not dividing n. The skeptical concern about a gap in the deformation-theoretic step therefore does not land. The main theorems are precisely stated and the central reconstruction argument is internally consistent.

minor comments (8)
  1. [Section 10.3] The assertions about the range 4≤n≤9, in particular the statement that β is an isomorphism for every field of characteristic not dividing n when 7≤n≤9, are presented with only a sketch and with reference to [8]. These claims are outside the main theorem, so they do not affect Theorems 1.1 and 1.2, but the wording 'It follows easily that β is in fact an isomorphism for every field' should be clearly marked as a summary of Borisov–Roulleau's computations or supplied with a full proof, so that the reader can distinguish new results from survey claims.
  2. [Lemma 15.5] The displayed exact sequence for restriction to a plane cubic has the wrong twist: for a cubic divisor C⊂P^2_A the kernel is O_{P^2_A}(−3), not O_{P^2_A}(−2). The conclusion is unaffected because both H^0 and H^1 vanish for either twist, so this is a typographical error rather than a gap, but it should be corrected.
  3. [Section 10.3] The reference '(Theorem 4.3)' in the second paragraph of Section 10.3 appears to be a misprint; the unique cubic through the ten-point window is proved in Corollary 4.3, and the numbering should be corrected.
  4. [Lemma 6.1] There is a missing space and period in the sentence 'entries of this grid are pairwise distinct in Z/nZEight of the nine entries lie in the window'; it should read 'in Z/nZ. Eight of the nine entries...'.
  5. [Remark 13.1] The display showing the sum over i of [P^i] is malformed ('n−1X i=0'); please correct the summation notation for readability.
  6. [Introduction, Section 1.1] The sentence 'We do not work with Z[1/n]-schemes, rather than Z-schemes' is grammatically incomplete; it should read 'We work with Z[1/n]-schemes rather than Z-schemes...'.
  7. [Appendix A and Section 11] The notation R_M is used both for the ordinary realization scheme and for the band scheme in Appendix A. Please disambiguate these objects (for example, by writing R_M^{band} for the band scheme) to avoid confusion in statements such as Proposition A.10 and Definition A.9.
  8. [Proposition 12.6] The coordinate-recovery argument at the end of the proof is quite terse, especially the sentence explaining that the Laurent monomial in the pinned units 'corrects' a displayed ratio to multidegree zero. Spelling out this correction explicitly for at least one coordinate, such as Y_i when B_i={i,1,2}, would improve readability without adding much length.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the reconstructions in Theorems 1.1 and 1.2 are self-contained; prior results are used only as external moduli-theoretic input or for noncentral small cases.

full rationale

I walked the paper's derivation chain and found no step that reduces to its own inputs by construction. The field-valued correspondence for n≥10 is proved directly from projective geometry: an arbitrary realization of T_n is fed through the seed pencil of Corollary 4.3, Chasles propagation (Lemmas 5.2, 6.2, 6.3), irreducibility/smoothness (Section 7), and the group-law recovery (Lemma 8.2 and Propositions 8.3–8.4). No parameter is fitted and no Borisov–Roulleau input is used in this range; the cases n=10 and n=11 are handled in Sections 10.1–10.2. The scheme-theoretic theorem is bootstrapped from the field-valued bijection through the Artinian-point criterion (Proposition 14.1), with the Artinian reconstruction proved independently via relative seed pencils, relative Chasles (Lemma 15.3), and relative Picard-theoretic group-law recovery (Lemmas 15.5–15.6). The external inputs are Deligne–Rapoport, Katz–Mazur, Conrad, EGA, and Stacks Project results on moduli of generalized elliptic curves and étaleness criteria; these do not include the target theorem. The paper's self-citations (Baker–Lorscheid foundations, Baker–Jin–Lorscheid band schemes, and the forthcoming [3]) occur in the appendix and in contextual remarks, not in the proof of the main theorems. The stated limitations—failure for n≤6, the citable Borisov–Roulleau treatment for 7≤n≤9, and the exclusion of primes dividing n in Remark 13.1—are explicit and do not conceal a circular dependence. The equivalence with Mazur's theorem is a genuine biconditional rather than a derivation of Mazur's theorem from itself.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims rest on standard theorems in algebraic geometry and matroid theory, plus one unproved computational assertion about small n. No fitted parameters are introduced, and no new physical or mathematical entities are postulated to make the argument work. The choice of basis triples B_i in Section 12 is a presentation choice, not a fitted quantity.

assumptions (7)
  • standard math Representability and smoothness of X_1(n) over Z[1/n] and existence of the universal generalized elliptic curve (Deligne-Rapoport, Katz-Mazur, Conrad)
    Invoked in Section 13.1 as background; used to define X_1(n)^∘ and to construct the modular morphism beta.
  • domain assumption The relative complete linear system |3O| embeds any irreducible generalized elliptic curve (smooth or Neron 1-gon) in P^2 and satisfies O_C(1) isomorphic to O_C(3O)
    Used in Sections 3 and 13.2 for marked plane cubics and the universal family; cited to Conrad [9] but not proved in the paper.
  • standard math Chasles and Cayley-Bacharach theorem
    Theorem 4.4 is the key propagation tool; a standard result from Eisenbud-Green-Harris [11].
  • standard math Group law on the smooth locus of an irreducible plane cubic; cuspidal case gives G_a and has no n-torsion when char(k) not dividing n
    Used in Sections 8 and 9; cited to Artin-Rodriguez-Villegas-Tate [1].
  • standard math Partial fields over fields are perfect, so weak and strong Grassmann-Plucker functions coincide
    Used in Lemma A.7 for the band scheme; cited to Baker-Bowler [2].
  • standard math Artinian-point criterion for isomorphisms of locally finitely presented morphisms between Noetherian schemes
    Proposition 14.1, proved in the paper using EGA IV 17.14.2 and Stacks Project references.
  • ad hoc to paper For 4 at most n at most 6, the k-realization space of T_n is a single point; for 7 at most n at most 9, it is P^1 minus 3, 4, or 5 points
    Asserted in Section 10.3 to follow by direct computation, but the computation is not displayed; used only in the small-n discussion, not in the main n at least 10 theorem.

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Cite this review

Pith. "Pith review of Elliptic matroids and modular curves." pith.science (2026). https://pith.science/paper/6NYEXJIX

@misc{pith2026260805299,
  author       = {Pith},
  title        = {Pith review of: Elliptic matroids and modular curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6NYEXJIX}},
  note         = {Machine review of arXiv:2608.05299}
}
abstract

For $n\geq 4$, let $T_n$ be the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets. Let $X_1(n)^\circ$ denote the open subscheme of the modular curve $X_1(n)$ obtained by removing the cusps corresponding to reducible N\'eron polygons. For $n \geq 10$, we give a purely algebraic and incidence-theoretic proof that, for every field $k$ with $\mathrm{char}(k)$ not dividing $n$, there is a natural bijection between $X_1(n)^\circ(k)$ and rescaling classes of $k$-realizations of $T_n$. For $k = \mathbb{C}$, this recovers a theorem of Borisov and Roulleau. We then upgrade the field-valued correspondence to an isomorphism of schemes over $\mathbb{Z}[1/n]$. The main new ingredient is a deformation-theoretic argument which allows us to verify the isomorphism on points valued in Artinian local rings. As consequences, the modular curve $X_1(n)^\circ$ acquires a natural model over $\mathbb{Z}[1/n]$ as a matroid realization space, and, for primes $p \geq 11$, the non-representability of $T_p$ over $\mathbb{Q}$ is equivalent to the prime-order case of Mazur's celebrated theorem on rational torsion points of elliptic curves. In an appendix, we explain how to upgrade the realization space of a matroid from an affine scheme over $\mathbb{Z}$ to an affine band scheme (in the sense of Baker-Jin-Lorscheid) over $\mathbb{F}_1^{\pm}$.

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Works this paper leans on

17 extracted references · 15 canonical work pages

  1. [1]

    Artin, F

    M. Artin, F. Rodriguez-Villegas, and J. Tate,On the Jacobians of plane cubics, Adv. Math.198(2005), no. 1, 366–382

    work page 2005

  2. [2]

    Baker and N

    M. Baker and N. Bowler,Matroids over partial hyperstructures, Adv. Math.343(2019), 821–863

    work page 2019

  3. [3]

    Baker and G

    M. Baker and G. Edwards and N. Solomon,Foundations of elliptic matroids, in preparation

    work page

  4. [4]

    Baker, T

    M. Baker, T. Jin, and O. Lorscheid,New building blocks forF 1-geometry: bands and band schemes, J. Lond. Math. Soc. (2)111(2025), no. 4, Paper No. e70125

    work page 2025

  5. [5]

    Baker and O

    M. Baker and O. Lorscheid,Foundations of matroids, Part 1: Matroids without large uniform minors, Mem. Amer. Math. Soc.305(2025), no. 1536, v+84 pp

    work page 2025

  6. [6]

    Baker and O

    M. Baker and O. Lorscheid,On a theorem of Lafforgue, Int. Math. Res. Not. IMRN 2024, no. 15, 11082–11091

    work page 2024

  7. [7]

    Baker and O

    M. Baker and O. Lorscheid,The moduli space of matroids, Adv. Math.390(2021), Paper No. 107883

    work page 2021

  8. [8]

    Borisov and X

    L. Borisov and X. Roulleau,Modular curvesX 1(n)as moduli spaces of point arrangements and applications, preprint, 2024, arXiv:2404.04364

    arXiv 2024

Show all 17 references
  1. [9]

    Conrad,Arithmetic moduli of generalized elliptic curves, J

    B. Conrad,Arithmetic moduli of generalized elliptic curves, J. Inst. Math. Jussieu6(2007), no. 2, 209–278

    2007

  2. [10]

    Deligne and M

    P. Deligne and M. Rapoport,Les sch´ emas de modules de courbes elliptiques, inModular Functions of One Variable II, Lecture Notes in Math. 349, Springer, 1973, 143–316

    1973

  3. [11]

    Eisenbud, M

    D. Eisenbud, M. Green, and J. Harris,Cayley–Bacharach theorems and conjectures, Bull. Amer. Math. Soc. (N.S.)33(1996), no. 3, 295–324

    1996

  4. [12]

    Grothendieck and J

    A. Grothendieck and J. Dieudonn´ e, ´El´ ements de g´ eom´ etrie alg´ ebrique IV:´Etude locale des sch´ emas et des morphismes de sch´ emas, Quatri` eme partie, Inst. Hautes´Etudes Sci. Publ. Math. No. 32 (1967)

    1967

  5. [13]

    Green and T

    B. Green and T. Tao,On sets defining few ordinary lines, Discrete Comput. Geom.50(2013), no. 2, 409–468

    2013

  6. [14]

    Katz and B

    N. Katz and B. Mazur,Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies 108, Princeton University Press, 1985

    1985

  7. [15]

    Mazur,Modular curves and the Eisenstein ideal, Inst

    B. Mazur,Modular curves and the Eisenstein ideal, Inst. Hautes ´Etudes Sci. Publ. Math. No. 47 (1977), 33–186

    1977

  8. [16]

    The Stacks Project Authors,The Stacks Project,https://stacks.math.columbia.edu

  9. [17]

    N. L. White,The bracket ring of a combinatorial geometry. I, Trans. Amer. Math. Soc. 202 (1975), 79–95

    1975

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