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REVIEW 3 major objections 4 minor 15 references

On Sun's Conjectures for Truncated Jacobi-Symbol Determinants via Supersingular Elliptic Curves

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Pith's one-line read Through exact diagonalization of the residue-class matrix and Deuring's reduction theorem, this paper proves Sun's conjectures on truncated Jacobi-symbol determinants: eight families of vanishings and divisibilities, plus n^2-divisibility…

desk verdict A genuinely new unified proof of Sun's truncated-determinant conjectures, with a concrete and localized gap in the exact-zero cases over F_p that needs a quadratic-extension repair before the paper is final. read the letter →

arxiv 2608.08509 v1 pith:C6PPX354 submitted 2026-08-09 math.NT

Guo Li , Xiaoju Yan This is my paper · ORCID
classification math.NT MSC 11A0711E2511G05
keywords Jacobi-symboldeterminantstruncatedLegendresymbolssupersingularellipticcurvescomplexmultiplicationHasseinvariantsDeuringreductiontheoremSunconjectures
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

The paper proves a large share of Zhi-Wei Sun's conjectures about truncated Jacobi-symbol determinants {c,d}_n = det[( ($j^{2}$+cjk+$dk^{2}$)/n )]_{2≤j,k≤n-2}. Theorem 1.1 settles eight conjecture families: exact vanishing or p-, $p^{2}$-, and $n^{2}$-divisibility for the parameter pairs (3,2), (2,2), (4,2), (8,8), (3,3), (42,-7), (21,112), (2,3), (6,15), (5,5), (10,9), and (8,18) under the stated congruence conditions on n, and Theorem 1.3 shows $n^{2}$ divides the full nonzero-residue determinants (2,3)_n and (6,15)_n for every odd n>3. The engine is a single reduction: diagonalizing the matrix indexed by F_p^× produces eigenvalues that are coefficients of q(X)^{(p-1)/2} with q(X)=$X^{2}$+cX+d, and the vanishing of those coefficients is shown to be the supersingularity of an explicit CM elliptic curve via Deuring's theorem. If the paper is right, scattered determinant identities become corollaries of one geometric criterion, and the same template generates further families.

What carries the argument

The load-bearing object is the exact eigenvalue decomposition of M^×_p(c,d) on the power-vector basis, with eigenvalues -a_s, together with the symmetry a_{p-1-s}=$d^{{s-(p-1)/2}}$a_s. It converts every determinant statement into a statement about coefficients of q(X)^{(p-1)/2}. The coefficient vanishing is then carried by the Hasse invariant Hap(E)=[$x^{{p-1}}$]f(x)^{(p-1)/2} of an explicitly given elliptic curve: supersingular reduction implies the invariant is zero, and Deuring's reduction theorem supplies the supersingularity when p is inert in the curve's CM field. The final piece is the kernel-vector transfer: zero eigenvalues at reciprocal indices produce a vector in the kernel of the truncated matrix, and Lemma 2.4's tensor-product argument lifts these local kernels to composite moduli n.

What would settle it

Compute {2,2}_13 directly as an integer from the 10×10 matrix with entries (($j^{2}$+2jk+$2k^{2}$)/13), 2≤j,k≤11. Theorem 1.1(b) predicts the determinant is exactly 0; any nonzero value would disprove the claim. Similarly, {2,2}_23 is predicted to be divisible by 23, and the paper's own Remark 3.6 gives the value, so a direct check settling that divisibility would test the same chain.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is a chain of equivalences linking combinatorics to arithmetic geometry. For an odd prime p with p∤d, the (p-1)×(p-1) residue matrix M^×_p(c,d) is diagonalized by the power vectors w_s=(1^s,2^s,...,(p-1)^s)^T, whose eigenvalues are -a_s where q(X)^{(p-1)/2}=∑_{s=0}^{p-1}a_s X^s in F_p[X]; the coefficient symmetry a_{p-1-s}=$d^{{s-(p-1)/2}}$a_s doubles each vanishing. The truncated determinant {c,d}_p is then forced to vanish or be divisible by p when two reciprocal coefficients vanish, because the difference of the corresponding kernel vectors vanishes at the deleted indices 1 and -1; for exact zero over Z, a character-coset indicator vector and an absolute-value bound do the job. Each coefficient a_s is identified with a Hasse invariant of an explicit elliptic curve—$y^{2}$=x($x^{2}$+cx+d) for the middle coefficient, $E_T^{{(3)}}$ and $E_T^{{(4)}}$ for the one-third and one-quarter coefficients—and computation of j-invariants reveals complex multiplication by orders of discriminants -4, -8, -3, -28, -7, -24, -20, and -6. Deuring's reduction theorem then converts inertness of p in the CM field into supersingularity, which makes the Hasse invariant zero, and the Chinese Remainder Theorem lifts the resulting kernel vectors from primes to squarefree composite moduli. This is the proof of Theorem 1.1 and Theorem 1.3.

Load-bearing premise

The proof leans on two previously published congruences of Z.-H. Sun that identify the one-third and one-quarter Legendre-polynomial values with elliptic-curve Hasse invariants; if either congruence failed for a prime in the stated residue classes, the vanishing results in parts (b), (f), (g), and (h) of Theorem 1.1 would no longer follow.

Editorial extensions

If this is right

  • For every odd n≡1 (mod 4) that is not a sum of two squares, {3,2}_n=0; this is part (a) of Theorem 1.1.
  • For the one-third family, {2,2}_p=0 for primes p≡13,19 (mod 24) and {2,2}_p≡0 (mod p) for p≡17,23 (mod 24), so the parity of the kernel-vector exponents decides between exact vanishing and mere p-divisibility.
  • For composite squarefree n, the CRT tensor-product lemma turns any local kernel modulo a prime divisor into a kernel of the truncated matrix modulo n, yielding n^2-divisibility for (2,3)_n and (6,15)_n and for {2,3}_n whenever n>3 and n not ±1 mod 12.
  • The same three-step template stated in Remark 3.11 predicts new explicit families, including p|{22,125}_p for p≡5,11 (mod 12) and p^2|{22,125}_p for p≡17,53 (mod 60).

Reading between the lines

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

  • One could run the same three-step template systematically: enumerate coefficient degrees s for which a Legendre-polynomial congruence is known, solve the corresponding (c,d) parameter equations, compute the CM field of the resulting curve, and read off new determinant congruences; the one-sixth example in Remark 3.11 is the first such output.
  • For forms of higher degree than X^2+cX+d, the analogous eigenvalues would be coefficients of f(X)^{(p-1)/2} with f of higher degree, and the Hasse invariant would be replaced by the corresponding invariant of the Jacobian or abelian variety, using the generalized Deuring theorem cited at the end of the paper.
  • The exact-zero mechanism suggests a statistical picture: a pair (c,d) should exhibit vanishing whenever the associated CM curve is supersingular at all primes in a dense-enough set, so the set of moduli n with {c,d}_n=0 should be governed by the splitting behaviour of primes in the CM field.
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Editorial analysis

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a unified proof of several conjectures of Z.-W. Sun on truncated Jacobi-symbol determinants {c,d}_n. The method diagonalizes the nonzero-residue matrix M^×_p(c,d) over F_p, identifies its eigenvalues with the coefficients a_s of q(X)^{(p-1)/2}, and then proves the required coefficient vanishings by linking them to Hasse invariants of explicitly given CM elliptic curves. Deuring's reduction theorem converts inertness of p in the relevant CM field into supersingularity, which forces the coefficient to vanish; composite moduli are handled by a Chinese-remainder tensor-product argument. The main results are Theorem 1.1, covering Sun's conjectures in eight families, and Theorem 1.3, giving n^2-divisibility for the full nonzero-residue determinants (2,3)_n and (6,15)_n.

Significance. If the proof can be repaired, this is a substantial contribution: it resolves a substantial batch of Sun's conjectures through a single conceptual mechanism, connecting truncated Legendre-symbol determinants to supersingular reduction and complex multiplication. The linear algebra in Lemmas 2.1–2.3 is clean and explicit, the coefficient-to-Hasse-invariant translations are concrete, and the included Sage listings provide machine-checked verification of the CM data against Hilbert class polynomials. The low-degree coefficient arguments in §3.4 and the central-coefficient arguments in §3.1 are checkable and persuasive. The paper is also honest about its external inputs, namely Z.-H. Sun's congruences and Deuring's theorem, and it does not assume or fit any target result; the conjectures serve as inputs, not parameters.

major comments (3)
  1. [§3.2, Eq. (6); §3.3, Eq. (10); proof of Theorem 1.1(b),(f),(h)] The proofs of the exact-zero cases choose α∈F_p with α^2=d, but for the primes on which those conclusions depend this α does not exist. For (c,d)=(2,2), p≡13,19 (mod 24) satisfies (2/p)=-1; for (c,d)=(5,5), p≡13,17 (mod 20) satisfies (5/p)=-1; and for (c,d)=(8,18), p≡13 (mod 24) satisfies (18/p)=-1. Consequently equations (6) and (10) are not legitimate as written, and the identities a_r=0 on which Theorem 1.1(b), (f), and (h) rest are not established. The good-reduction discussion in §3.2 shows awareness that the parameter T may live in a quadratic extension, but the Legendre-polynomial and Hasse-invariant computation is not reworked over F_{p^2}. A repair must rewrite (6)/(10), Corollaries 3.5 and 3.8, and the Deuring step for reductions whose residue field is F_{p^2}. Direct computation for p=13 supports the truth of the conclusion, but the manuscript's internal derivation is invalid for these primes as it stands.
  2. [§3.2 after Eq. (9); Theorem 3.2] Even after moving the computation to F_{p^2}, the implication 'E_t/F_{p^2} is supersingular ⇒ H_p^{(3)}(t)=0' is not immediate from Theorem 3.2 as stated. Theorem 3.2(1) is phrased for a curve over F_p, while the reduction E_t in the non-residue cases is defined over F_{p^2}. The missing bridge is the identity Ha_{p^2}(E)=H_p(E)^{p+1} for these cubics, which follows by extracting the coefficient of x^{p^2-1} in f^{(p^2-1)/2}=f^{(p-1)/2}(f^{(p-1)/2})^p. This identity would show that supersingularity over F_{p^2} forces the ordinary p-th Hasse invariant to vanish. The same issue affects the one-quarter coefficient in Corollary 3.8. The paper should state and prove this Hasse-invariant relation explicitly, since it is load-bearing for the exact-zero and p^2-divisibility conclusions.
  3. [§3.2–§3.3, Theorems 3.4 and 3.7] The one-third and one-quarter coefficient vanishings import Z.-H. Sun's congruences without proof. As external published results this is acceptable in principle, but the application to t=-α^{-1} with α outside F_p is not covered by the stated hypotheses: those theorems are for t∈Z_(p), and the Legendre-symbol sums involve g_t(x) or f_t(x) evaluated in F_p. A citation cannot fix the mismatch. The authors should either verify that Corollaries 3.5 and 3.8 remain valid, as polynomial identities over F_p[T], at F_{p^2}-valued t, or supply a direct extension of the congruences to the quadratic residue field. Until this is done, parts (b), (f), (g), and (h) rest on black-box hypotheses that are not met at the primes used in the exact arguments.
minor comments (4)
  1. [Table 1] The table is poorly formatted; in the (3,3) row the columns for Δ(E) and j(E) appear merged as '−2 433 0−3', and the factorization is not readable. Please reformat so that each column is unambiguous.
  2. [Remark 3.11] The remark states additional families, such as {22,125}_p, {28,250}_p, and {126,2625}_p, with proofs omitted. If these are intended as results, they should be proved or explicitly labelled as conjectural; as written, an unproved assertion in a remark is not a contribution.
  3. [Acknowledgments] The acknowledgment mentions 'GPT-5.6 Sol' as an assistant. This is not a mathematical defect, but the paper should either specify precisely how the tool was used or omit the mention, to avoid ambiguity about the provenance of the mathematical arguments.
  4. [§3.1, exceptional prime p=3] In the discussion before the proof of Theorem 1.1(a), the vector u_3=(0,1,-1)^T is described as a kernel vector 'over Z'; it would be clearer to note explicitly that its entries are integers and that it vanishes at the boundary coordinates 0,1,-1, which is what allows Lemma 2.4(ii) to be applied.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Sun's conjectures are treated as targets, not assumptions, and all load-bearing ingredients are independently sourced or verified.

full rationale

The paper does not assume any of the determinant conjectures it claims to prove; those conjectures are the output of the derivation. The derivation chain is genuinely external to the target statements: the eigenvalues of the nonzero-residue matrix are computed as explicit coefficients a_s of q(X)^{(p-1)/2}; these coefficients are identified with Hasse invariants of explicitly written elliptic curves; Deuring's reduction theorem is invoked from standard literature (Lang, Silverman, Zaytsev); and the claimed CM fields are verified separately by exact Hilbert class polynomial computations in SageMath (Appendix A, Listings 1-3), which is an independent benchmark not depending on the determinant results. The imported Legendre-polynomial congruences of Z.-H. Sun (Theorems 3.4 and 3.7) are external theorems with proofs elsewhere, not self-citations, and they are used as tools rather than as disguised restatements of the target. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the present authors, and no known result is merely relabeled. There is an apparent algebraic gap in Sections 3.2 and 3.3, where the text chooses alpha in F_p with alpha^2 = d for d equal to 2, 5, or 18 in residue classes where d is a quadratic nonresidue; this affects the validity of equations (6) and (10) as written and is a correctness concern, but it is not circularity because no target conclusion is used to establish itself. Accordingly, the circularity score is 0.

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

The account does not fit free parameters to the determinant values: every c,d pair is fixed by the conjecture, every t is an algebraic expression, and the CM discriminants are read off from Hilbert class polynomials. The central claim rests on external theorems, namely Deuring reduction and two Legendre-polynomial congruences by Z.-H. Sun, and on the correctness of the Sage computations. No new entities are postulated.

assumptions (7)
  • domain assumption Deuring's reduction theorem: for a CM elliptic curve with good reduction, the reduction is supersingular iff p is inert or ramified in the CM field.
    Invoked as Theorem 3.1 in Section 3 and is the main bridge from inertness conditions to vanishing coefficients. Cited to Lang and Zaytsev, not proved in this paper.
  • domain assumption Z.-H. Sun's one-third Legendre-polynomial congruence (Ref [9], Theorem 3.1).
    Used as Theorem 3.4 in Section 3.2 to turn P_{floor(p/3)}(t) into a Hasse invariant. The proof is not reproduced.
  • domain assumption Z.-H. Sun's one-quarter Legendre-polynomial congruence (Ref [7], Theorem 2.1).
    Used as Theorem 3.7 in Section 3.3 for P_{floor(p/4)}(t); imported without proof.
  • standard math Euler criterion and finite-field power-sum identity.
    Converts Legendre symbol entries into coefficients of q(X)^((p-1)/2) in Lemma 2.1; standard.
  • standard math Hasse bound and trace-supersingularity equivalence.
    Theorem 3.2 from Silverman; identifies the central coefficient as the Hasse invariant and gives t_p=0 for supersingular curves with p>=5.
  • standard math CRT tensor decomposition of Jacobi-symbol matrices on squarefree moduli.
    Lemma 2.4; depends on Jacobi multiplicativity and the Chinese remainder theorem.
  • domain assumption The j-invariants in Tables 1 and 2 have the CM discriminants claimed, as verified by SageMath.
    Appendix A provides executable Sage scripts; the review did not independently recompute. Any error in this data would break individual cases.

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Pith. "Pith review of On Sun's Conjectures for Truncated Jacobi-Symbol Determinants via Supersingular Elliptic Curves." pith.science (2026). https://pith.science/paper/C6PPX354

@misc{pith2026260808509,
  author       = {Pith},
  title        = {Pith review of: On Sun's Conjectures for Truncated Jacobi-Symbol Determinants via Supersingular Elliptic Curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C6PPX354}},
  note         = {Machine review of arXiv:2608.08509}
}
abstract

We study the truncated Jacobi-symbol determinants $$\{c,d\}_n=\det\!\left[\left(\frac{j^2+cjk+dk^2}{n}\right)\right]_{2\le j,k\le n-2}$$ proposed by Zhi-Wei Sun in \cite{ZWSun} and prove Conjectures 5.1(i), 5.2, 5.3, 5.4, 5.5, 5.6(i), 5.7, 5.8, a case of 5.6(ii) of \cite{ZWSun}, Conjecture 4.8(i) of \cite{Sun2019} and some strengthened forms. All results follow from a single unified approach: for a prime $p$, we diagonalize the nonzero-residue matrix indexed by $\mathbb{F}_p^\times$ and reduce the vanishing of determinants to the supersingular reduction of certain CM elliptic curves. The same framework extends naturally to further families of parameters, suggesting a general mechanism behind identities of this type.

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Works this paper leans on

15 extracted references · 9 canonical work pages

  1. [1]

    Carlitz, Some cyclotomic matrices, Acta Arith.5(1959), 293–308

    L. Carlitz, Some cyclotomic matrices, Acta Arith.5(1959), 293–308. doi:10.4064/aa-5-3-293-308

    doi:10.4064/aa-5-3-293-308 1959

  2. [2]

    R. J. Chapman, Determinants of Legendre symbol matrices, Acta Arith.115(2004), no. 3, 231–244, doi:10.4064/aa115-3-4

    doi:10.4064/aa115-3-4 2004

  3. [3]

    R. J. Chapman, My evil determinant problem, preprint, available at http://empslocal.ex.ac.uk/people/staff/rjchapma/etc/evildet.pdf, 2012

    work page 2012

  4. [4]

    On some determinants involving Jacobi symbols

    D. Krachun, F. Petrov, Z.-W. Sun, and M. Vsemirnov, On some determinants involving Jacobi symbols,Finite Fields Appl.64(2020), Article 101672, arXiv:1812.08080

    work page Pith review arXiv 2020

  5. [5]

    Lang,Elliptic Functions, second ed., Graduate Texts in Mathematics 112, Springer-Verlag, 1987

    S. Lang,Elliptic Functions, second ed., Graduate Texts in Mathematics 112, Springer-Verlag, 1987

    work page 1987

  6. [6]

    J. H. Silverman,The Arithmetic of Elliptic Curves, second ed., Graduate Texts in Mathematics 106, Springer, 2009

    work page 2009

  7. [7]

    Sun, Congruences concerning Legendre polynomials II,Journal of Number Theory, 133 (2013), no

    Z.-H. Sun, Congruences concerning Legendre polynomials II,Journal of Number Theory, 133 (2013), no. 6, 1950–1976

    work page 2013

  8. [8]

    Congruences concerning Legendre polynomials III

    Z.-H. Sun, Congruences concerning Legendre polynomials III, arXiv:1012.4234v4, 2012

    work page Pith review arXiv 2012

Show all 15 references
  1. [9]

    Sun, Congruences involving 2k k 23k k m−k,Journal of Number Theory, 133 (2013), no

    Z.-H. Sun, Congruences involving 2k k 23k k m−k,Journal of Number Theory, 133 (2013), no. 5, 1572–1595

    2013

  2. [10]

    Sun, On some determinants with Legendre symbol entries,Finite Fields Appl.56(2019), 285–307, doi:10.1016/j.ffa.2018.12.004

    Z.-W. Sun, On some determinants with Legendre symbol entries,Finite Fields Appl.56(2019), 285–307, doi:10.1016/j.ffa.2018.12.004

    2019 doi

  3. [11]

    Sun, Problems and results on determinants involving Legendre symbols, arXiv:2405.03626v8, 2024

    Z.-W. Sun, Problems and results on determinants involving Legendre symbols, arXiv:2405.03626v8, 2024

    2024 arXiv

  4. [12]

    Vsemirnov, On the evaluation of R

    M. Vsemirnov, On the evaluation of R. Chapman’s ”evil determinant”, Linear Algebra Appl.436(2012), no. 11, 4101–4106, 10.1016/j.laa.2011.08.039

    2012 doi

  5. [13]

    Vsemirnov, On R

    M. Vsemirnov, On R. Chapman’s ”evil determinant”: casep≡1 (mod 4), Acta Arith.159(2013), no. 4, 331–344, 10.4064/aa159-4-3. 14

    2013 doi

  6. [14]

    Wu, Elliptic curves overF p and determinants of Legendre matrices, arXiv:2012.05746, 2021

    H.-L. Wu, Elliptic curves overF p and determinants of Legendre matrices, arXiv:2012.05746, 2021

    2012 arXiv

  7. [15]

    Zaytsev, Generalization of Deuring reduction theorem, J

    A. Zaytsev, Generalization of Deuring reduction theorem, J. Algebra392(2013), 97–114 doi:10.1016/j.jalgebra.2013.06.017. School of Mathematical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China Email address:liguo22@mails.ucas.ac.c...

    2013 doi

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