There are infinitely many elliptic curves over the rationals of rank 2

David Zywina

REVIEW 2 major objections 5 minor 1 cited by

There are infinitely many elliptic curves over the rationals of rank 2

T0 review · 2 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper proves that infinitely many elliptic curves over the rationals have rank exactly 2 by an explicit 2-descent.

desk verdict A genuinely careful 2-descent producing an explicit infinite family of rank-2 curves, with a repairable but real arithmetic error in the final j-invariant step. read the letter →

arxiv 2502.01957 v1 pith:YUUH4DIU submitted 2025-02-04 math.NT

David Zywina This is my paper · ORCID

classification math.NT MSC 11G1814J27

keywords ellipticcurvesrank2rationalpoints2-descentSelmergroupssimultaneousprimesj-invariant

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 infinitely many elliptic curves over the rationals have Mordell-Weil rank exactly 2, not merely at least 2. The curves come from an explicit two-parameter family $y^{2}$ = $x^{3}$ - 5(m+$16n^{2}$)$x^{2}$ + 4(m+$16n^{2}$)(m+$25n^{2}$)x, where m, m+$16n^{2}$ and m+$25n^{2}$ are primes congruent to 11 modulo 24. A 2-descent along a degree-2 isogeny computes the relevant Selmer groups, showing E(Q)/2E(Q) is isomorphic to (Z/2Z)^3 and that the rational torsion subgroup has order 2, so the rank is exactly 2. The infinitude of such prime triples is imported from a cited theorem on simultaneous prime values of polynomials. This supplies the first rank r at least 2 that is confirmed to occur infinitely often among elliptic curves over Q.

What carries the argument

The argument runs on a 2-descent through an explicit degree-2 isogeny phi: E -> E' whose kernel is generated by the rational point (0,0). For each square class d, the phi-Selmer group is defined by local solubility of the curves $y^{2}$ = d $x^{4}$ + a' $x^{2}$ + b'/d, and the paper confines this Selmer group to four square classes using 2-adic, p-adic and Legendre-symbol arguments at the primes 2, m+$25n^{2}$ and m. The dual Selmer group is confined to four positive square classes using real solubility and solubility modulo m. Equality with the images of explicit rational points then forces E(Q)/2E(Q) to be (Z/2Z)^3, while the cited theorem on simultaneous prime values of polynomials guarantees infinitely many admissible input pairs. A final j-invariant calculation recovers the pair (m,n) from the denominator, proving that the resulting curves are distinct up to isomorphism over Q.

What would settle it

Take the smallest admissible pair (m,n) and compute E(Q)/2E(Q) by an explicit 2-descent or with a computer algebra system; if its size is not 8, or if E(Q) has a rational point of order 4, then Theorem 1.2 is false.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.2: whenever m and n are natural numbers with m, m+$16n^{2}$ and m+$25n^{2}$ primes congruent to 11 modulo 24, the curve $y^{2}$ = $x^{3}$ - 5(m+$16n^{2}$)$x^{2}$ + 4(m+$16n^{2}$)(m+$25n^{2}$)x satisfies E(Q) isomorphic to Z/2Z times $Z^{2}$. The points P0=(0,0), P1=(m+$16n^{2}$, 6n(m+$16n^{2}$)) and P2=($36n^{2}$, 12n(m-$2n^{2}$)) generate E(Q)/2E(Q), and the only rational torsion is the order-2 point P0. The proof computes the Selmer groups for a degree-2 isogeny and its dual, shows the natural inclusions into them are equalities, and concludes that E(Q)/2E(Q) has order 8. Different admissible pairs give different j-invariants, because the denominator of j isolates m and m+$25n^{2}$, so infinitely many prime triples from the cited theorem produce infinitely many non-isomorphic rank-2 curves.

Load-bearing premise

The load-bearing premise is that infinitely many pairs (m,n) exist for which m, m+$16n^{2}$ and m+$25n^{2}$ are all prime and congruent to 11 modulo 24; the paper takes this from a cited theorem and does not re-derive it.

Editorial extensions

If this is right

Every curve in the family has E(Q) exactly isomorphic to Z/2Z times Z^2, with explicit generators P0, P1 and P2 of E(Q)/2E(Q).
The j-invariant calculation shows that distinct admissible pairs yield distinct curves, so the rank-2 curves produced are provably non-isomorphic over Q.
A variant with m congruent to 5 modulo 24 yields infinitely many curves with root number -1 and rank 2 or 3, as the paper notes in its remarks.
Assuming the parity conjecture, that root-number -1 variant would give infinitely many elliptic curves over Q of rank 3.

Reading between the lines

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

Editorial inference: the same 2-isogeny descent should work for other square constants in place of 16 and 25, provided the corresponding Selmer solubility calculations and the local conditions of the cited prime-progressions theorem still hold.
Editorial inference: because the j-invariant denominator recovers (m,n), counting admissible prime triples with m and n up to X would give a quantitative lower bound for how many rank-2 curves the construction produces up to that height.
Editorial inference: a computer check of the smallest admissible pairs, verifying that E(Q)/2E(Q) has order 8 and that the torsion subgroup has order 2, would exercise the Selmer bounds and could reveal any hidden local obstruction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The 2-descent in this paper is the real thing. The Selmer computations in Lemmas 3.1–3.3 are detailed and coherent, and the exact sequence does force E(Q)/2E(Q) to be (Z/2Z)^3, so the rank-2 conclusion in Theorem 1.2 rests on solid ground. The explicit family and the three generating points are useful, and the use of Tao–Ziegler to get infinitely many admissible prime triples is the right idea. I would not call the result non-new just because [BJ16] and [Jeo19] have similar titles: the paper explicitly says those works are conditional on the parity conjecture, and if that is accurate, Theorem 1.1 is a genuine unconditional improvement. That claim should be stated in the introduction rather than buried in an aside, but it is not a fabrication.

The soft spot is in Section 4, and it is real. The stated j-invariant is wrong. For E: y^2 = x^3 - 5S x^2 + 4ST x with S = m+16n^2, T = m+25n^2, the standard invariants give c4 = 16S(13m+100n^2) and Delta = 2304 m S^3 T^2, hence j = 16(13m+100n^2)^3 / (9m T^2), not the paper's 16(...)^3/(32m T^2). The denominator in Lemma 4.1 should be 9mT^2, not 32mT^2, and the claimed gcd is false because both the numerator and the denominator contain a factor 16. The intended recovery of (m,n) from the largest prime divisors of the denominator still works with the corrected 9mT^2 denominator, since m and T are odd primes larger than 3 and do not divide 13m+100n^2. So the damage is contained, but as submitted the proof of Theorem 1.1 has a false computation at the final step.

A second, smaller gap: the paper simply cites [TZ08, Theorem 1.3] without checking the local admissibility conditions for the three polynomial forms. I expect they hold, but a referee should ask for a sentence or a short verification.

This paper is for elliptic curve specialists and anyone working on explicit rank families. It deserves refereeing, not desk rejection. I would send it back for revision with the Section 4 computation fixed and the Tao–Ziegler hypotheses made explicit.

Referee Report

2 major / 5 minor

Summary. The paper studies a two-parameter family of elliptic curves E_{m,n}: y^2 = x^3 - 5(m+16n^2)x^2 + 4(m+16n^2)(m+25n^2)x, where m, m+16n^2 and m+25n^2 are primes congruent to 11 modulo 24. A 2-descent via the rational 2-isogeny is used to show E(Q) ≅ Z/2Z × Z^2, giving rank exactly 2. The infinitude of such pairs is obtained from a Tao-Ziegler theorem, and a j-invariant calculation is supposed to show that distinct pairs give non-isomorphic curves over Q. Thus the paper claims infinitely many rank-2 elliptic curves over Q up to isomorphism over Q. The Selmer-group computations and the descent exact sequence appear coherent, but the j-invariant step contains a concrete arithmetic error that affects the proof of Theorem 1.1 as written.

Significance. If repaired, the paper gives a short, explicit, unconditional proof of the infinitude of rank-2 elliptic curves over Q within a specific family. The 2-descent computations are transparent, self-contained, and do not rely on BSD, the parity conjecture, or fitted parameters; the only external input is the cited Tao-Ziegler theorem. The main technical defect is localized to the j-invariant formula in Section 4 and is repairable by a straightforward correction. Assuming the corrected computation, the overall strategy is sound and the result is significant for a classical problem in arithmetic statistics.

major comments (2)

[§4, displayed jE and Lemma 4.1] The displayed formula for the j-invariant is incorrect. With S=m+16n^2 and T=m+25n^2, the invariants are c4=16S(13m+100n^2) and Δ=2304 m S^3 T^2, so jE=c4^3/Δ=16(13m+100n^2)^3/(9mT^2), not the stated denominator 32mT^2. Consequently Lemma 4.1 is false as stated: gcd(16(13m+100n^2)^3, 32mT^2) is at least 16 because both entries are divisible by 16. The proof only rules out divisibility by the odd primes 3, m and T and silently ignores the factor 2. Since the distinctness argument in Theorem 1.1 depends on identifying the denominator of jE in lowest terms, this computation is load-bearing. The error is repairable: replace 32 by 9, prove gcd((13m+100n^2)^3, 9mT^2)=1 using the already given odd-prime checks, and then the denominator 9mT^2 has largest prime divisors T and m, so the pair (m,n) is still recovered from jE.
[§4, first paragraph] The invocation of [TZ08, Theorem 1.3] does not verify the hypotheses of the theorem for the three forms m, m+16n^2 and m+25n^2 together with the residue class 11 mod 24. In particular, one should exhibit a local-admissibility reduction, for example by setting m=24a+11 and n=12b, so that the three forms become 24a+11, 24(a+96b^2)+11 and 24(a+150b^2)+11, which have no fixed prime divisor; one should also note that the theorem gives infinitely many points with n>0. Please add this short verification; because the Tao-Ziegler theorem is the only source of infinitude in Theorem 1.1, this deserves an explicit check rather than a one-line citation.

minor comments (5)

[§4, unnumbered sentence before Lemma 4.1] The sentence 'jE uniquely determines E up to isomorphism over Q' is false: over Q, equal j-invariants determine a curve only up to quadratic twist. The distinctness argument only needs the implication that Q-isomorphic curves have the same j, so the conclusion is unaffected; please rephrase the sentence.
[Lemma 3.2, proof] The first sentence of the proof refers to 'Sel_φ(E'/Q)' but the lemma concerns Sel_{\hat φ}(E'/Q); the φ should be \hat φ.
[Lemma 3.3, part (ii)] In part (ii) the notation δ(P1) should be δ'(P1), since the map from E(Q) to Q^×/(Q^×)^2 is denoted δ' in Section 2.
[Abstract and Theorem 1.1] The abstract states 'up to isomorphism over \bar Q' while Theorem 1.1 states 'up to isomorphism over Q'; these two statements should be aligned.
[§1.1] The displayed symbol '/CG (E/Q)[2]' appears to be a typesetting corruption of the Tate-Shafarevich group; please fix it in both occurrences.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: rank statement from self-contained 2-descent; Tao–Ziegler cited as external independent input; self-citation only motivational.

full rationale

The paper's central claim, Theorem 1.2, is proven by a direct 2-descent computation: the Selmer groups are bounded by local arguments in Lemmas 3.1 and 3.2, equality with the images of the explicit points is shown in Lemma 3.3, and the rank 2 conclusion follows from Lemmas 3.4 and 3.5 together with the short exact sequence for 2-isogeny descent. No parameter is fitted to any elliptic curve data and no rank assumption is smuggled into the descent; the argument is self-contained given the congruence hypotheses on m, m+16n^2 and m+25n^2. The infinitude in Theorem 1.1 rests on the Tao–Ziegler polynomial Szemerédi theorem, cited as an external black box that guarantees infinitely many admissible prime triples; the paper does not derive that theorem from its own results, and neither the theorem nor its hypotheses involve the rank of the constructed curves. The sequel [Zyw25] is cited only in Section 1.1 as motivation and is explicitly marked as not used elsewhere, so it is not load-bearing. The skeptical observation that the j-invariant formula in Section 4 appears to have a missing factor of 9 in the denominator is a possible correctness defect in the distinctness argument, not a circularity: recovering (m,n) from the denominator is an independent verification step, and an algebraic error cannot make a derivation circular. Thus no circular step is exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new theoretical entities and fits no constants. Its assumptions beyond standard descent theory are the external Tao-Ziegler theorem and standard results about torsion and j-invariants.

assumptions (4)

standard math Standard 2-isogeny descent theory, including Selmer group definitions and Lemma 2.1's bound that squarefree d dividing b' represents every Selmer class.
Invoked throughout Section 2 and used without proof; standard from Silverman.
standard math Tate's algorithm and the formal group description of torsion over Q_2 (Silverman, Propositions VII.2.1 and IV.3.2, and Algorithm 9.4).
Used in Lemma 3.4 to bound the rational torsion subgroup to a 2-group.
domain assumption Tao-Ziegler polynomial Szemeredi theorem provides infinitely many simultaneous prime values for m, m+16n^2, m+25n^2 in the 11 mod 24 congruence class.
Section 4 cites [TZ08, Theorem 1.3] but does not verify the admissibility and local conditions for this specific triple of polynomials.
standard math The expression for the j-invariant and the fact that the j-invariant determines the curve up to isomorphism over the algebraic closure.
Used in Section 4 to distinguish curves and recover the pair (m,n); standard.

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Pith. "Pith review of There are infinitely many elliptic curves over the rationals of rank 2." pith.science (2026). https://pith.science/paper/YUUH4DIU

@misc{pith2026250201957,
  author       = {Pith},
  title        = {Pith review of: There are infinitely many elliptic curves over the rationals of rank 2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YUUH4DIU}},
  note         = {Machine review of arXiv:2502.01957}
}

abstract

We show that there are infinitely many elliptic curves $E/\mathbb{Q}$, up to isomorphism over $\overline{\mathbb{Q}}$, for which the finitely generated group $E(\mathbb{Q})$ has rank exactly $2$. Our elliptic curves are given by explicit models and their rank is shown to be $2$ via a $2$-descent. That there are infinitely many such elliptic curves makes use of a theorem of Tao and Ziegler.

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Cited by 1 Pith paper

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Elliptic curves of rank one over number fields
math.NT 2025-05 conditional novelty 6.0 of 10

For every number field K and every 3-generic elliptic curve E/K with full rational 2-torsion, there are infinitely many quadratic twists of E with rank exactly 1.

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Works this paper leans on

2 extracted references · 1 canonical work pages · cited by 1 Pith paper

[1]

Japan Acad

[BJ16] Dongho Byeon and Keunyoung Jeong, Inﬁnitely many elliptic curves of rank exactly two , Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 5, 64–66, DOI 10.3792/pjaa.92.64. MR3492814 ↑1.1 [Jeo19] Keunyoung Jeong, Inﬁnitely many elliptic curves of rank exactly two II , Proc. Japan Acad. Ser. A Math. Sci. 95 (2019), no. 6, 53–57, DOI 10.3792/pjaa.95.5...

work page doi:10.3792/pjaa.92.64 2016
[1994]

Reine Angew

MR1312368 ↑3.2 [Sil83] , Heights and the specialization map for families of abelian v arieties, J. Reine Angew. Math. 342 (1983), 197–211, DOI 10.1515/crll.1983.342.197. MR703488 ↑1, 1.1 [TZ08] Terence Tao and Tamar Ziegler, The primes contain arbitrarily long polynomial progressio ns, Acta Math. 201 (2008), no. 2, 213–305, DOI 10.1007/s11511-008-0032-5. ...

arXiv 1983

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Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Japan Acad

[BJ16] Dongho Byeon and Keunyoung Jeong, Inﬁnitely many elliptic curves of rank exactly two , Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 5, 64–66, DOI 10.3792/pjaa.92.64. MR3492814 ↑1.1 [Jeo19] Keunyoung Jeong, Inﬁnitely many elliptic curves of rank exactly two II , Proc. Japan Acad. Ser. A Math. Sci. 95 (2019), no. 6, 53–57, DOI 10.3792/pjaa.95.5...

work page doi:10.3792/pjaa.92.64 2016

[2] [1994]

Reine Angew

MR1312368 ↑3.2 [Sil83] , Heights and the specialization map for families of abelian v arieties, J. Reine Angew. Math. 342 (1983), 197–211, DOI 10.1515/crll.1983.342.197. MR703488 ↑1, 1.1 [TZ08] Terence Tao and Tamar Ziegler, The primes contain arbitrarily long polynomial progressio ns, Acta Math. 201 (2008), no. 2, 213–305, DOI 10.1007/s11511-008-0032-5. ...

arXiv 1983