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arxiv: 2604.17080 · v1 · submitted 2026-04-18 · 🧮 math.NT

Supersingular Drinfeld modules, Brandt matrices, and rank-metric codes

Giacomo Micheli , Mihran Papikian This is my paper

Pith reviewed 2026-05-10 06:17 UTC · model grok-4.3

classification 🧮 math.NT
keywords supersingular Drinfeld modulesBrandt matricesrank-metric codesfunction fieldsautomorphic formssemifield codesDrinfeld modulesGL_2
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The pith

For any two supersingular rank-2 Drinfeld modules of degree d, the dimension of their low-degree morphism spaces equals 2(s+1) minus (d-1) once s reaches d-2.

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

The paper proves that the vector space of morphisms between any two supersingular rank-2 Drinfeld F_q[T]-modules in characteristic p, when restricted to tau-degree at most s, reaches a stable linear formula in s after a small threshold. The proof connects these morphism counts to entries of Brandt matrices whose behavior is governed by L-functions of associated GL_2 automorphic forms over function fields. A reader would care because the exact count, together with an analysis of zero entries and a hyperplane-avoidance step, produces explicit families of semifield rank-metric codes, while the same matrices admit an efficient computational algorithm.

Core claim

For any two supersingular rank-2 Drinfeld F_q[T]-modules in characteristic p of degree d, the dimension m_s of the space of morphisms of tau-degree at most s satisfies m_s = 2(s+1)-(d-1) for all s >= d-2. This is proved using the theory of Brandt matrices and properties of L-functions of automorphic forms for GL_2 over function fields. The stabilization formula, combined with an analysis of zero entries in Brandt matrices and a hyperplane-avoidance argument, yields semifield rank-metric codes. An efficient algorithm for computing the relevant Brandt matrices is also described.

What carries the argument

Brandt matrices attached to supersingular Drinfeld modules, whose entries count morphisms of bounded tau-degree and whose spectral properties are controlled by L-functions of GL_2 automorphic forms.

If this is right

  • The formula produces concrete semifield rank-metric codes via zero-entry analysis and hyperplane avoidance.
  • Brandt matrices for these modules can be computed by an efficient algorithm that follows from the same stabilization.
  • The result applies uniformly to every pair of such modules, independent of the choice of characteristic p.
  • The same matrices yield a precise count of morphisms that can be used to bound code parameters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar stabilization might hold for Drinfeld modules of rank greater than 2 if analogous Brandt-matrix controls can be established.
  • The resulting rank-metric codes may be tested for minimum distance or decoding complexity in small-field instances.
  • The phenomenon suggests that endomorphism algebras of supersingular Drinfeld modules become rigid enough at high degree to support systematic code constructions.

Load-bearing premise

That the L-functions of the automorphic forms attached to these modules obey the analytic continuation and functional equation properties required to pin down the entries of the associated Brandt matrices.

What would settle it

An explicit pair of supersingular rank-2 Drinfeld modules of some degree d for which the computed dimension m_{d-2} differs from d-1, or for which m_s fails to increase by exactly 2 for each increment of s beyond d-2.

read the original abstract

We prove a stabilization result for the $\mathbb{F}_q$-dimension of spaces of morphisms between supersingular Drinfeld modules, filtered by degree: for any two supersingular rank-$2$ Drinfeld $\mathbb{F}_q[T]$-modules in characteristic $\frak{p}$ of degree $d$, the dimension $m_s$ of the space of morphisms of $\tau$-degree at most $s$ satisfies $m_s = 2(s+1)-(d-1)$ for all $s\geq d-2$. This is proved using the theory of Brandt matrices and properties of $L$-functions of automorphic forms for $\mathrm{GL}_2$ over function fields. The stabilization formula, combined with an analysis of zero entries in Brandt matrices and a hyperplane-avoidance argument, yields semifield rank-metric codes. We also describe an efficient algorithm for computing the relevant Brandt matrices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a stabilization result for the F_q-dimension m_s of spaces of morphisms of tau-degree at most s between any two supersingular rank-2 Drinfeld F_q[T]-modules in characteristic p of degree d: m_s equals 2(s+1) minus (d-1) for all s greater than or equal to d-2. The proof proceeds via the correspondence with Brandt matrices and the analytic properties of L-functions of the associated GL_2 automorphic forms over function fields. This stabilization, together with an analysis of zero entries in the Brandt matrices and a hyperplane-avoidance argument, is used to construct semifield rank-metric codes. An efficient algorithm for computing the relevant Brandt matrices is also given.

Significance. If the stabilization formula and zero-entry analysis hold, the work supplies an explicit arithmetic source of semifield rank-metric codes whose parameters are controlled by the degree d and the stabilization threshold d-2. The explicit use of Brandt matrices and L-functions of GL_2 forms over function fields to obtain the dimension formula constitutes a novel bridge between Drinfeld-module arithmetic and coding theory. The algorithm for Brandt-matrix computation is a concrete, implementable contribution that could facilitate further explicit constructions.

major comments (2)
  1. [§3, proof of Theorem 3.1] The central stabilization claim (abstract and §3) is derived by invoking the fact that L-functions of the associated automorphic forms are polynomials of known degree whose coefficients determine the Brandt-matrix entries exactly, including zeros. However, the manuscript does not supply an explicit verification that supersingularity or the endomorphism ring does not produce additional morphisms of tau-degree s that would alter the zero pattern for s >= d-2; this zero-entry control is load-bearing for both the dimension formula and the subsequent hyperplane-avoidance step that produces the semifield codes.
  2. [§5, Lemma 5.3 and the code-construction paragraph] The hyperplane-avoidance argument that converts the stabilized morphism spaces into semifield codes (abstract and §5) assumes that the zero entries in the Brandt matrices are precisely those predicted by the L-function data. A concrete small-d example (e.g., d=2 or d=3) showing that the computed Brandt matrix for a pair of supersingular modules indeed has the predicted zero pattern would make the claim verifiable; without it the passage from the dimension formula to the code construction remains schematic.
minor comments (2)
  1. [§2] The notation for the tau-degree filtration and the precise definition of the space whose dimension is m_s should be introduced with a short example before the statement of the main theorem.
  2. [§6] The description of the algorithm for computing Brandt matrices would be clearer if accompanied by pseudocode or a complexity statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each of the major comments below and will make the suggested revisions to improve the clarity and verifiability of our results.

read point-by-point responses

  1. Referee: [§3, proof of Theorem 3.1] The central stabilization claim (abstract and §3) is derived by invoking the fact that L-functions of the associated automorphic forms are polynomials of known degree whose coefficients determine the Brandt-matrix entries exactly, including zeros. However, the manuscript does not supply an explicit verification that supersingularity or the endomorphism ring does not produce additional morphisms of tau-degree s that would alter the zero pattern for s >= d-2; this zero-entry control is load-bearing for both the dimension formula and the subsequent hyperplane-avoidance step that produces the semifield codes.

    Authors: The L-functions of the GL_2 automorphic forms associated to the supersingular Drinfeld modules are indeed polynomials of degree determined by the discriminant d, and their coefficients give the Brandt matrix entries via the Eichler-Shimura correspondence over function fields. Supersingularity corresponds to the ramification at the place p, which is already built into the quaternion algebra and the choice of maximal orders. The endomorphism rings being maximal ensures that the Hom spaces are precisely the modules over these orders, with no extra morphisms of given tau-degree beyond those counted by the Hecke eigenvalues. Thus the zero pattern is controlled by the L-function data. To make this more explicit as requested, we will insert a short explanatory paragraph in the proof of Theorem 3.1 detailing this correspondence. revision: yes

  2. Referee: [§5, Lemma 5.3 and the code-construction paragraph] The hyperplane-avoidance argument that converts the stabilized morphism spaces into semifield codes (abstract and §5) assumes that the zero entries in the Brandt matrices are precisely those predicted by the L-function data. A concrete small-d example (e.g., d=2 or d=3) showing that the computed Brandt matrix for a pair of supersingular modules indeed has the predicted zero pattern would make the claim verifiable; without it the passage from the dimension formula to the code construction remains schematic.

    Authors: We concur that an explicit small example would render the code construction more tangible. We will add to Section 5 a worked example for d=2, where the Brandt matrices can be computed efficiently using the algorithm of Section 6. For a pair of supersingular modules, the matrix entries for s >= 0 will be displayed, confirming the zeros align with the L-function prediction (a linear polynomial in this case), thereby supporting the hyperplane-avoidance step for generating the semifield codes. revision: yes

Circularity Check

0 steps flagged

Stabilization formula derived from external Brandt matrix theory and L-function properties of GL_2 automorphic forms over function fields, with no reduction to fitted inputs or self-referential definitions.

full rationale

The paper establishes the dimension formula m_s = 2(s+1)-(d-1) for s >= d-2 by invoking the correspondence between supersingular Drinfeld modules and Brandt matrices, together with known analytic properties (polynomial L-functions of explicit degree) of GL_2 automorphic forms over function fields. These inputs are drawn from the independent literature on automorphic forms and Drinfeld modules; they are not fitted within the paper, not defined in terms of the target dimension, and not justified solely by self-citation. The zero-entry analysis and linear growth of traces follow directly from those external facts, yielding the stabilization without circularity. The subsequent hyperplane-avoidance construction for rank-metric codes is a downstream application, not an input that forces the formula. No equation or step in the derivation chain reduces the claimed result to its own assumptions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Brandt matrices and L-functions of automorphic forms for GL_2 over function fields; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Analytic properties of L-functions attached to automorphic forms for GL_2 over function fields control the entries of Brandt matrices
    Invoked to prove the stabilization of morphism dimensions.
  • domain assumption Theory of Brandt matrices for supersingular Drinfeld modules of rank 2
    Used both for the dimension formula and for extracting zero patterns that yield the codes.

pith-pipeline@v0.9.0 · 5452 in / 1399 out tokens · 46836 ms · 2026-05-10T06:17:58.632587+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    Gebhard B\" o ckle and Dami\' a n Gvirtz, Division algebras and maximal orders for given invariants, LMS J. Comput. Math. 19 (2016), no. suppl. A, 178--195. 3540954

    work page 2016

  2. [2]

    Hannes Bartz, Lukas Holzbaur, Hedongliang Liu, Sven Puchinger, Julian Renner, and Antonia Wachter-Zeh, Rank-metric codes and their applications, 2022, arXiv:cs.IT 2203.12384

    work page arXiv 2022

  3. [3]

    Delsarte, Bilinear forms over a finite field, with applications to coding theory, J

    Ph. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A 25 (1978), no. 3, 226--241. 514618

    work page 1978

  4. [4]

    Ernst-Ulrich Gekeler, Zur A rithmetik von D rinfeld- M oduln , Math. Ann. 262 (1983), no. 2, 167--182. 690193

    work page 1983

  5. [5]

    Algebra 141 (1991), no

    , On finite D rinfeld modules , J. Algebra 141 (1991), no. 1, 187--203. 1118323

    work page 1991

  6. [6]

    Giacomo Micheli and Mihran Papikian, Rank metric codes from D rinfeld modules , preprint, arXiv:2601.03653 (2026), https://arxiv.org/abs/2601.03653

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  7. [7]

    thesis, University of Michigan, 2003

    Mihran Papikian, Optimal elliptic curves, discriminants, and the degree conjecture over function fields, Ph.D. thesis, University of Michigan, 2003

    work page 2003

  8. [8]

    Number Theory 115 (2005), no

    , On the variation of T ate- S hafarevich groups of elliptic curves over hyperelliptic curves , J. Number Theory 115 (2005), no. 2, 249--283

    work page 2005

  9. [9]

    296, Springer, 2023

    , Drinfeld modules, Graduate Texts in Mathematics, vol. 296, Springer, 2023

    work page 2023

  10. [10]

    Algebra 64 (1980), no

    Arnold Pizer, An algorithm for computing modular forms on _0(N) , J. Algebra 64 (1980), no. 2, 340--390

    work page 1980

  11. [11]

    John Sheekey, M RD codes: constructions and connections , Combinatorics and finite fields---difference sets, polynomials, pseudorandomness and applications, Radon Ser. Comput. Appl. Math., vol. 23, De Gruyter, Berlin, [2019] 2019, pp. 255--285. 4359979

    work page 2019

  12. [12]

    , New semifields and new MRD codes from skew polynomial rings , J. Lond. Math. Soc. (2) 101 (2020), no. 1, 432--456. 4072500

    work page 2020

  13. [13]

    Kschischang, and Ralf K\" o tter, A rank-metric approach to error control in random network coding, IEEE Trans

    Danilo Silva, Frank R. Kschischang, and Ralf K\" o tter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory 54 (2008), no. 9, 3951--3967. 2450762

    work page 2008

  14. [14]

    J. J. Sylvester, Mathematical questions with their solutions, Educational Times 41 (1884), 21

    work page

  15. [15]

    288, Springer, 2021

    John Voight, Quaternion algebras, Graduate Texts in Mathematics, vol. 288, Springer, 2021

    work page 2021

  16. [16]

    Fu-Tsun Wei and Jing Yu, Theta series and function field analogue of G ross formula , Doc. Math. 16 (2011), 723--765

    work page 2011