2-Selmer groups, 2-class groups, and congruent numbers
Pith reviewed 2026-05-08 05:32 UTC · model grok-4.3
The pith
If n factored from primes ≡5 mod 8 and one ≡7 mod 8 is a congruent number, then the class number h(−n) of Q(√−n) must be divisible by a power of 2 that grows with the number of factors.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
If n = p1 p2 ⋯ pt q with each pi ≡5 mod 8 and q ≡7 mod 8 is a congruent number, then the class number h(−n) of Q(√−n) satisfies a specific divisibility condition. Quantitative lower bounds are given on the number of non-congruent n of this form. For n = p1⋯pt q with q ≡3 mod 8, if n is congruent then the class numbers of Q(√−n) and Q(√−p1⋯pt) are congruent modulo a power of 2.
What carries the argument
The relation between the 2-Selmer group of the elliptic curve y² = x³ − n²x and the 2-class group of the imaginary quadratic field Q(√−n).
Load-bearing premise
The 2-Selmer group of the congruent-number curve is controlled by the 2-class group of Q(√−n) in exactly the way needed to turn Selmer-rank data into class-number divisibility for these residue classes mod 8.
What would settle it
A concrete square-free n of the stated form that is a congruent number yet whose class number h(−n) has 2-adic valuation strictly smaller than the predicted lower bound.
read the original abstract
In this article, we study necessary conditions for certain square-free integers to be congruent numbers. Our method uses divisibility properties of class numbers of related imaginary quadratic fields. We first consider positive square-free integers of the form $n = p_1 p_2 \cdots p_t q,$ where each prime $p_i \equiv 5 \pmod{8}$ and $q \equiv 7 \pmod{8}$. We show that if such an integer $n$ is a congruent number, then the class number $h(-n)$ of the quadratic field $\mathbb{Q}(\sqrt{-n})$ satisfies a specific divisibility condition. Furthermore, we provide quantitative lower bounds on the number of non-congruent numbers of this form. Next, we study integers of the form $n = p_1 p_2 \cdots p_t q,$ with $p_i \equiv 5 \pmod{8}$ and $q \equiv 3 \pmod{8}$. Assuming that $n$ is a congruent number, we obtain a congruence modulo powers of $2$ between the class numbers of the fields $\mathbb{Q}(\sqrt{-n})$ and $\mathbb{Q}\!\left(\sqrt{-p_1 p_2 \cdots p_t}\right)$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies necessary conditions for square-free n of the form product of t primes ≡5 mod 8 times a prime q (either ≡7 or ≡3 mod 8) to be congruent numbers. It claims that if such an n is congruent then the class number h(−n) of Q(√−n) satisfies a specific 2-power divisibility (for q≡7 mod 8), gives quantitative lower bounds on the count of non-congruent n in this family, and (for q≡3 mod 8) establishes a congruence modulo 2^m between h(−n) and the class number of Q(√−p1⋯pt). The arguments proceed via 2-Selmer groups of the congruent-number curve E_n : y² = x³ − n²x and their relation to 2-class groups.
Significance. If the central claims hold, the work supplies explicit arithmetic constraints linking the 2-Selmer rank of E_n to the 2-rank of Cl(Q(√−n)) for these residue classes, together with density-type lower bounds on non-congruent numbers. Such results could be useful for computational verification of the congruent-number problem and for refining heuristics that relate ranks of congruent-number curves to class numbers of imaginary quadratic fields.
major comments (3)
- [Derivation of the divisibility condition for q≡7 mod 8] The central translation from 2-Selmer rank of E_n to a lower bound on the 2-rank of Cl(Q(√−n)) (used for the divisibility claim when q≡7 mod 8) requires that the natural map from the class group via homogeneous spaces is surjective onto the Selmer group with cokernel independent of the specific splitting at 2 and the primes p_i, q. For primes ≡5 or 7 mod 8 the local conditions at those primes and the 2-adic place differ from the generic case; an explicit verification that no extra 2-torsion survives in the Selmer group (or that Sha[2] contributes trivially) is needed, as otherwise the implied lower bound on h(−n) can fail even when rank(E_n(Q))>0.
- [Section giving quantitative lower bounds] The quantitative lower bounds on the number of non-congruent n of the stated form rest directly on the above divisibility; if the 2-Selmer-to-class-group control does not hold uniformly for the chosen residue classes, the counting argument requires adjustment or additional local-obstruction analysis.
- [Treatment of the case q≡3 mod 8] The congruence modulo 2^m between h(−n) and h(−p1⋯pt) (for q≡3 mod 8) likewise assumes the same precise control of the 2-Selmer group; the paper must confirm that the local Hilbert symbols at 2 and the primes do not introduce cokernel elements that would break the stated congruence.
minor comments (2)
- [Abstract] The abstract refers to “a specific divisibility condition” without stating the precise power of 2; including the exact statement would aid readability.
- [Notation and setup] Notation for the 2-part of the class number versus the full class number h(−n) should be clarified throughout, especially when relating Selmer ranks to class-group ranks.
Simulated Author's Rebuttal
We thank the referee for their careful reading and valuable comments, which help clarify the necessary verifications for our arguments relating 2-Selmer groups to class groups in these specific arithmetic families. We address each major comment point by point below and will revise the manuscript to incorporate explicit local computations where needed.
read point-by-point responses
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Referee: [Derivation of the divisibility condition for q≡7 mod 8] The central translation from 2-Selmer rank of E_n to a lower bound on the 2-rank of Cl(Q(√−n)) (used for the divisibility claim when q≡7 mod 8) requires that the natural map from the class group via homogeneous spaces is surjective onto the Selmer group with cokernel independent of the specific splitting at 2 and the primes p_i, q. For primes ≡5 or 7 mod 8 the local conditions at those primes and the 2-adic place differ from the generic case; an explicit verification that no extra 2-torsion survives in the Selmer group (or that Sha[2] contributes trivially) is needed, as otherwise the implied lower bound on h(−n) can fail even when rank(E_n(Q))>0.
Authors: We agree that the surjectivity of the natural map from the 2-class group to the 2-Selmer group of E_n, along with control of its cokernel, must be verified explicitly for the residue classes p_i ≡5 mod 8 and q ≡7 mod 8, as the local conditions at 2 and these primes deviate from generic cases. The manuscript employs the standard 2-descent via homogeneous spaces for congruent-number curves, but we acknowledge that a dedicated case analysis is required to rule out extra 2-torsion or nontrivial Sha[2] contributions. In the revision, we will add a new subsection computing the relevant local Hilbert symbols at 2 and the primes in these classes, confirming that the cokernel is trivial (or constant) and that the lower bound on the 2-rank of Cl(Q(√−n)) holds precisely when rank(E_n(Q)) > 0. revision: yes
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Referee: [Section giving quantitative lower bounds] The quantitative lower bounds on the number of non-congruent n of the stated form rest directly on the above divisibility; if the 2-Selmer-to-class-group control does not hold uniformly for the chosen residue classes, the counting argument requires adjustment or additional local-obstruction analysis.
Authors: The quantitative lower bounds on non-congruent n in this family are derived directly from the divisibility condition on h(−n). With the explicit local verification added as outlined in response to the first comment, the control of the 2-Selmer group will be uniform across the family, and no adjustment to the counting argument will be needed. For completeness, we will also include a short paragraph discussing the absence of additional local obstructions at the primes p_i and q that could affect the density estimates. revision: partial
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Referee: [Treatment of the case q≡3 mod 8] The congruence modulo 2^m between h(−n) and h(−p1⋯pt) (for q≡3 mod 8) likewise assumes the same precise control of the 2-Selmer group; the paper must confirm that the local Hilbert symbols at 2 and the primes do not introduce cokernel elements that would break the stated congruence.
Authors: We agree that the stated congruence modulo 2^m between h(−n) and h(−p1⋯pt) relies on precise control of the 2-Selmer group, including that local Hilbert symbols at 2 and the primes do not introduce extra cokernel elements. The manuscript uses the relation between the Selmer group of E_n and the class group of Q(√−p1⋯pt) when q ≡3 mod 8, but we will add an explicit computation of these local symbols in the revision to confirm the congruence holds without additional 2-torsion contributions. This verification will parallel the one for the q ≡7 mod 8 case and ensure the result is rigorous. revision: yes
Circularity Check
No significant circularity; derivations rely on standard Selmer-class group relations
full rationale
The paper establishes necessary conditions for certain n to be congruent numbers by linking 2-Selmer ranks of the curve y² = x³ - n²x to 2-ranks of class groups of Q(√-n), then derives divisibility statements on h(-n) and lower bounds on non-congruent n. These steps invoke established properties of 2-descent and homogeneous spaces rather than redefining quantities in terms of themselves or fitting parameters to the target conclusions. No equations or citations reduce the central mapping to a tautology, and the quantitative counting arguments for non-congruent numbers are independent of the core translation. The work is self-contained against external benchmarks in algebraic number theory.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The 2-Selmer rank of y² = x³ − n²x equals the 2-rank of the class group of Q(√−n) plus a correction term controlled for the residue classes mod 8
- standard math Standard properties of the 2-class group and its relation to the narrow class group in real quadratic fields
Reference graph
Works this paper leans on
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[1]
,The size of Selmer groups for the congruent number problem. II, Invent. Math., 118 (1994), pp. 331–370. With an appendix by P. Monsky. [14]K. Heegner,Diophantische analysis und modulfunktionen, Mathematische Zeitschrift, 56 (1952), pp. 227–253. [15]H. Jung and Q. Yue,8-ranks of class groups of imaginary quadratic number fields and their densities, J. Kor...
work page 1994
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[2]
,Congruent numbers, quadratic forms andK2, Math. Ann., 383 (2022), pp. 1647–1686. [20]J. H. Sil verman,The arithmetic of elliptic curves, vol. 106 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1986. [21]W. Stein et al.,Sage Mathematics Software (Version 9.3), The Sage Group, 2024. [22]J. B. Tunnell,A classical diophantine problem and modula...
work page 2022
discussion (0)
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