{"id":"ddeb6106-0b63-4744-9b29-dad1e47004c8","arxiv_id":"2502.01957","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For m, m+16n^2, 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 has rank exactly 2, giving infinitely many rank-2 curves.","lead":"This paper constructs an explicit infinite family of elliptic curves over the rational numbers and proves by a 2-descent computation that every curve in the family has rank exactly 2. It is a clean arithmetic-geometry result, though the paper's own citations suggest the main theorem may already be known.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"j-invariant formula in §4 is incorrect, invalidating the distinctness step of Theorem 1.1 as written; Lemma 4.1's gcd claim is false.","rationale":"The reader's weakest assumption focused on the Tao-Ziegler application. That step is underexplained but likely sound: TZ08 gives infinitely many polynomial progressions in primes, and restricting to the arithmetic progression 11 mod 24 is a standard positive-density subset corollary; local admissibility holds for the configuration. I therefore do not regard TZ08 as the most load-bearing gap. Instead, the paper contains a provably incorrect j-invariant formula and a false gcd lemma in the final step of Theorem 1.1. Because Theorem 1.1 asserts infinitude up to isomorphism, the distinctness of j-invariants is essential. The error is concrete and can be fixed by replacing 32 by 9, after which the recovery of (m,n) from the denominator's largest prime divisors works. The reader did flag the false gcd claim, but not the underlying j-invariant mistake. This reinforces the CONDITIONAL verdict: the central theorem is plausible and the argument is repairable, but the manuscript as written contains a real computational error in a load-bearing step. No change to the reader's verdict is needed.","tokens_in":10084,"tokens_out":36193,"duration_ms":317012,"concrete_test":"Recompute the j-invariant from the Weierstrass equation y^2 = x^3 - 5S x^2 + 4ST x using j = 256(a^2 - 3b)^3 / (a^2 b^2 - 4b^3) with a = -5S, b = 4ST, and compare with the formula in §4. If the resulting denominator is 9mT^2 instead of 32mT^2, the paper's formula is confirmed incorrect. Then check Lemma 4.1 numerically with m=11, n=1: the gcd is at least 16, refuting the stated equality.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The final step of Theorem 1.1 shows that distinct pairs (m,n) give non-isomorphic curves by claiming the j-invariant is jE = 16(13m+100n^2)^3 / (32m(m+25n^2)^2) and that this is in lowest terms. This formula 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 = 16(a^2 - 3b) = 16S(13m+100n^2) and Δ = 16b^2(a^2 - 4b) = 2304 m S^3 T^2, so j = c4^3/Δ = 16(13m+100n^2)^3 / (9m(m+25n^2)^2), not the paper's denominator 32m. The error propagates to Lemma 4.1, which claims gcd(16(13m+100n^2)^3, 32m(m+25n^2)^2) = 1; the gcd is at least 16 since both sides have a factor 16. The paper's proof only checks the odd primes, ignoring the common factor 2. With the corrected denominator 9mT^2, the intended recovery of m and T from the largest prime divisors of the reduced denominator remains valid (since m and T are odd primes larger than 3 and do not divide 13m+100n^2, as the proof shows). Thus the defect is repairable, but as submitted the distinctness argument for infinitude up to isomorphism rests on a demonstrably false computation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10430,"tokens_out":27069,"duration_ms":260982,"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":[{"comment":"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.","section":"§4, displayed jE and Lemma 4.1"},{"comment":"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.","section":"§4, first paragraph"}],"minor_comments":[{"comment":"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.","section":"§4, unnumbered sentence before Lemma 4.1"},{"comment":"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.","section":"Lemma 3.2, proof"},{"comment":"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.","section":"Lemma 3.3, part (ii)"},{"comment":"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.","section":"Abstract and Theorem 1.1"},{"comment":"The displayed symbol '/CG (E/Q)[2]' appears to be a typesetting corruption of the Tate-Shafarevich group; please fix it in both occurrences.","section":"§1.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is short and the main descent computation appears sound. The j-invariant defect is localized and repairable, so major revision rather than rejection seems appropriate. I would also ask the author to spell out the Tao-Ziegler reduction; I do not see circularity or hidden parameter-fitting in the argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nA 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.\n\nThis 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.","headline":"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.","tokens_in":10990,"tokens_out":6343,"would_cite":true,"duration_ms":61535,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G18","14J27"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that infinitely many elliptic curves over the rationals have rank exactly 2 by an explicit 2-descent.","keywords":["elliptic curves","rank 2","rational points","2-descent","Selmer groups","simultaneous primes","j-invariant"],"falsifier":"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.","tokens_in":9847,"feed_emoji":"🔢","tokens_out":11418,"duration_ms":105062,"temperature":0.7,"pith_summary":"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.","feed_headline":"Infinitely many elliptic curves over the rationals have rank exactly 2","feed_subtitle":"It is the first rank above 1 proven to occur infinitely often among rational elliptic curves.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Its Theorem 1.3 supplies infinitely many pairs (m,n) with m, m+16n^2 and m+25n^2 simultaneously prime in the class 11 mod 24, which is the entire source of infinitude.","marker":"[TZ08]"},{"why":"Supplies the 2-isogeny descent formalism, the Selmer-group definition, and the torsion facts used in Lemmas 2.1, 3.3 and 3.4.","marker":"[Sil09]"},{"why":"Provides Algorithm 9.4, Tate's algorithm, used in Lemma 3.4 to determine the Kodaira type at 2 and bound the rational torsion.","marker":"[Sil94]"}],"fun_headline_variants":["Infinitely many rank-2 elliptic curves over Q","Rank 2 elliptic curves: infinite family via 2-descent","Explicit family proves infinite rank-2 elliptic curves","Tao-Ziegler primes give infinite rank-2 curves","Infinitely many elliptic curves reach rank 2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Infinitely many rank-2 elliptic curves over Q","Rank 2 elliptic curves: infinite family via 2-descent","Explicit family proves infinite rank-2 elliptic curves","Tao-Ziegler primes give infinite rank-2 curves","Infinitely many elliptic curves reach rank 2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000794,"raw_usage":{"total_tokens":3452,"prompt_tokens":858,"completion_tokens":2594,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":474,"completion_tokens_details":{"reasoning_tokens":2514}},"tokens_in":474,"tokens_out":2594,"duration_ms":18514,"temperature":1.0,"reasoning_tokens":2514,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T13:54:04.432395+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}