Monsky Matrix and 2-Selmer rank
Pith reviewed 2026-05-07 12:56 UTC · model grok-4.3
The pith
The Monsky matrix shows that the 2-Selmer rank is zero for infinite families of non-congruent numbers with many prime factors in residue classes 1, 2, and 3 modulo 8.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For families of n with prescribed prime-factor patterns and lying in the stated classes modulo 8, the Monsky matrix of the elliptic curve y^2 = x^3 - n^2 x has rank or determinant that makes the 2-Selmer group coincide with its known torsion part, so the 2-Selmer rank is zero; hence these n are non-congruent numbers, and quantitative counting shows each family is infinite.
What carries the argument
The Monsky matrix, whose size and rank determine the F_2-dimension of the 2-Selmer group of the congruent-number elliptic curve beyond its torsion.
If this is right
- There exist infinite families of non-congruent numbers in each of the residue classes 1, 2, and 3 modulo 8.
- Each such family contains numbers with arbitrarily large numbers of prime factors (in triples or quadruples).
- The 2-Selmer rank of the associated elliptic curve is exactly zero for every member of these families.
- Quantitative lower bounds on the number of non-congruent members in each family follow from the matrix computations.
Where Pith is reading between the lines
- The method supplies concrete, arbitrarily composite examples where the elliptic-curve rank is provably zero despite the presence of many prime factors in n.
- The same matrix technique could be tested on families with other residue conditions or with five or more prime factors to see whether the rank-zero conclusion persists.
- These families furnish test cases for conjectures on the average size of Selmer groups in thin sets of elliptic curves.
Load-bearing premise
The selected families of n, defined by their exact prime-factor counts and residue classes modulo 8, produce Monsky matrices whose kernels or determinants force the 2-Selmer group to contain nothing beyond the torsion subgroup.
What would settle it
An explicit n belonging to one of the families for which the elliptic curve y^2 = x^3 - n^2 x possesses a rational point of infinite order, or for which direct computation of its 2-Selmer group yields dimension greater than zero.
read the original abstract
In this article, we produce infinite families of non-congruent numbers in the residue class of $1,2,$ and $3$ modulo $8$ with arbitrarily many triples or quadruples prime factors. In short, we use Monsky matrix to show that the $2$-Selmer rank of the corresponding congruent number elliptic curve is zero. We also establish some quantitative results to conclude that each such family contains infinitely many non-congruent numbers.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript constructs infinite families of non-congruent numbers n ≡ 1, 2, or 3 mod 8, each having arbitrarily many prime factors arranged in triples or quadruples. It applies the Monsky matrix to prove that the 2-Selmer rank of the associated congruent-number elliptic curve E_n is zero (i.e., the Selmer group equals the torsion subgroup), and supplies quantitative estimates establishing that each family contains infinitely many such non-congruent numbers.
Significance. If the central claims hold, the work supplies new explicit infinite families of non-congruent numbers with controlled, arbitrarily large prime factorizations. This is of interest for the congruent-number problem and the distribution of 2-Selmer ranks of elliptic curves of the form y² = x³ − n²x. The approach relies on the standard Monsky matrix (whose entries are Legendre symbols together with local conditions at 2 and ∞), which is a recognized tool for bounding Selmer groups; credit is due for attempting to combine it with arithmetic-progression constructions that fix the necessary local conditions.
major comments (2)
- [§3 (family construction) and the proof of the main theorem] The construction of the families (via arithmetic progressions fixing residue classes mod 8 and selected Legendre symbols) does not include an explicit verification that the resulting Monsky matrix over F₂ has rank exactly equal to the number of odd prime factors when the number of primes grows arbitrarily. Quadratic reciprocity may introduce linear dependencies over F₂ among the rows/columns for large numbers of primes, which would enlarge the kernel and produce positive 2-Selmer rank, falsifying the zero-rank claim.
- [§4 (Monsky matrix application)] No sample computation of the Monsky matrix (with explicit entries and rank) is provided for even a modest member of any family (e.g., a product of three or four primes satisfying the stated conditions). Without such a check, it is impossible to confirm that the matrix forces dim Sel₂(E_n/Q) = 2 exactly.
minor comments (2)
- The abstract and introduction should clarify the precise prime-factorization pattern (e.g., exactly three or four primes, or products of such blocks) used to build the families.
- Notation for the Monsky matrix (its precise size, ordering of rows/columns, and the contribution of the 2-adic and real places) should be stated explicitly before the rank argument.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will revise the manuscript to incorporate explicit verifications and a sample computation.
read point-by-point responses
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Referee: [§3 (family construction) and the proof of the main theorem] The construction of the families (via arithmetic progressions fixing residue classes mod 8 and selected Legendre symbols) does not include an explicit verification that the resulting Monsky matrix over F₂ has rank exactly equal to the number of odd prime factors when the number of primes grows arbitrarily. Quadratic reciprocity may introduce linear dependencies over F₂ among the rows/columns for large numbers of primes, which would enlarge the kernel and produce positive 2-Selmer rank, falsifying the zero-rank claim.
Authors: The arithmetic progressions in §3 are selected to fix residue classes mod 8 together with the pairwise Legendre symbols (p_i/p_j) among the primes dividing n. These conditions are chosen so that the resulting Monsky matrix over F₂ is upper-triangular with nonzero diagonal entries (after accounting for the fixed local conditions at 2 and ∞), ensuring full rank equal to the number of odd prime factors. The reciprocity law does not introduce additional F₂-linear dependencies because the mod-8 classes fix the quadratic characters at 2 and the sign factors (-1)^{(p-1)/2(q-1)/2} uniformly for all primes in the family. Nevertheless, we agree that an explicit verification for arbitrarily large sets is not spelled out and will add a detailed argument (by induction on the number of primes, exhibiting the triangular form) in the revised §3. revision: yes
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Referee: [§4 (Monsky matrix application)] No sample computation of the Monsky matrix (with explicit entries and rank) is provided for even a modest member of any family (e.g., a product of three or four primes satisfying the stated conditions). Without such a check, it is impossible to confirm that the matrix forces dim Sel₂(E_n/Q) = 2 exactly.
Authors: We agree that a concrete example would make the argument more transparent. In the revised manuscript we will insert, in §4, an explicit sample computation for a number n that is a product of three primes satisfying the arithmetic-progression conditions of one of the families. The computation will list all Legendre-symbol entries of the Monsky matrix, incorporate the local conditions at 2 and ∞, compute its rank over F₂, and confirm that the rank equals the number of odd prime factors, yielding dim Sel₂(E_n/Q) = 2 as claimed. revision: yes
Circularity Check
No significant circularity detected
full rationale
The paper constructs explicit infinite families of integers n with prescribed prime-factor structure and residue classes modulo 8, then applies the standard Monsky matrix (whose entries are Legendre symbols at the primes together with local conditions at 2 and infinity) to conclude that the 2-Selmer rank of the associated congruent-number curve is zero. The Monsky matrix is an independently defined object from the literature and is not redefined or fitted inside the paper in terms of the target Selmer rank. No step reduces a claimed prediction to a fitted parameter or to a self-citation chain whose content is itself the result being proved; the infinitude statements rest on quantitative estimates that are separate from the rank computation. The derivation therefore contains independent mathematical content and is self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
Reference graph
Works this paper leans on
-
[1]
The parity of the rank of the Mordell-Weil group
[BS66] B. J. Birch and N. M. Stephens. “The parity of the rank of the Mordell-Weil group”. In:Topology5 (1966), pp. 295–299. [CG19] Weidong Cheng and Xuejun Guo. “Some new families of non-congruent numbers”. In:J. Number Theory 196 (2019), pp. 291–305. [DM26] Shamik Das and Sudipa Mondal. “A necessary condition for a congruent number of the form 8k+ 3”. I...
work page 1966
-
[2]
Families of even noncongruent numbers with arbitrarily many pairs of prime factors
[DS22] Shamik Das and Anupam Saikia. “Families of even noncongruent numbers with arbitrarily many pairs of prime factors”. In:Rocky Mountain J. Math.52.2 (2022), pp. 471–481. [DS24] Shamik Das and Anupam Saikia. “Families of non-congruent numbers with arbitrarily many triplets of prime factors”. In:Kyushu J. Math.78.1 (2024), pp. 119–128. 14 Table 1.Examp...
work page 2022
-
[3]
The size of Selmer groups for the congruent number problem. II
[HW79] G. H. Hardy and E. M. Wright.An introduction to the theory of numbers. Fifth. The Clarendon Press, Oxford University Press, New York, 1979, pp. xvi+426. [Hea94] D. R. Heath-Brown. “The size of Selmer groups for the congruent number problem. II”. In:Invent. Math. 118.2 (1994). With an appendix by P. Monsky, pp. 331–370. [Kim+24] Ringi Kim, Junguk Le...
work page 1979
-
[4]
Secr´ etariat Math., Paris, 1975, Exp. No. 16,
work page 1975
-
[5]
Diophantine equations, class groups and non-congruent numbers
[LQ23] Guilin Li and Hourong Qin. “Diophantine equations, class groups and non-congruent numbers”. In:Ra- manujan J.62.4 (2023), pp. 1081–1105. [LQ24] Guilin Li and Hourong Qin. “The Monsky matrices and non-congruent numbers”. In:Algebra Colloq.31.2 (2024), pp. 239–262. [Mey00] Carl Meyer.Matrix analysis and applied linear algebra. With 1 CD-ROM (Windows,...
work page 2023
-
[6]
On non-congruent numbers as multiples of non-congruent numbers
Graduate Texts in Mathematics. Springer- Verlag, New York, 1986, pp. xii+400. [Zha26] Shenxing Zhang. “On non-congruent numbers as multiples of non-congruent numbers”. In:J. Algebra687 (2026), pp. 394–418. Email address:shamikd@iitk.ac.in Email address:sudipa.mondal123@gmail.com 15
work page 1986
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