{"id":"891b36dd-2048-45dc-90e8-21ee0c846197","arxiv_id":"2502.01026","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For the elliptic curve E_t: y^2 = x(x^2 - x + t), the paper proves that E_t(Q) has rank 0 for infinitely many rational t.","lead":"David Zywina proves the first known elliptic curve family over Q(T) where infinitely many rational specializations have rank equal to the generic rank, namely zero. He uses a theorem on arithmetic progressions of primes to build specializations with few bad primes, then a 2-descent shows the rank is zero.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.5's Cassels/Dokchitser period-ratio step is the load-bearing point: if the minimal-model normalization of Ω_E/Ω_E' or the |c| ∈ {1,1/2} dichotomy is applied incorrectly, the bound |Sel_φ̂| = 2 collapses and the rank-zero conclusion fails.","rationale":"I read the proof in good faith and found no explicit mathematical error. Lemma 2.4 is a careful and correct local-solvability argument; Lemma 2.6 correctly reduces rank zero to the bound |Sel_2| ≤ 2; the use of Green's theorem to produce infinitely many admissible triples is standard and the injection from Silverman specialization correctly forces rank E(Q(T)) = 0. The single place where a hidden assumption could overturn the central claim is Lemma 2.5, exactly as the reader flagged. The multiplier c for the standard 2-isogeny between the stated minimal models is indeed 1/2, not 1, so the proof appears internally consistent, but the paper does not verify the hypotheses of [DD15] or give a computational cross-check. A direct Selmer computation for one admissible triple would settle whether the period-ratio and Tamagawa factors are correct, and would not require rerunning the entire descent. Because no concrete flaw has been identified, I do not recommend changing the reader's ACCEPT verdict; the proposed check is a verification step that would raise confidence from moderate to high.","tokens_in":8393,"tokens_out":57435,"duration_ms":545762,"concrete_test":"For the smallest admissible triple (m,n) = (3,8), with primes 3, 11, 19 all ≡ 3 mod 8, compute with Magma or Sage the full 2-Selmer group and rank of E: y^2 = x^3 - 36x^2 + 2376x. Also compute Sel_φ and Sel_φ̂ directly, and verify |Sel_φ̂| = 2. Independently compute the real periods of the minimal models of E and E' (with E' given by y^2 = x^3 + 18x^2 - 513x) and check Ω_E/Ω_E' = 1/2, together with the Tamagawa ratios from Lemmas 2.2 and 2.3. If the computed Selmer orders or the period ratio differ, Lemma 2.5 has a normalization error; if they match, the load-bearing step is confirmed for the family.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The rank-zero conclusion depends entirely on Lemma 2.5, which imports its numerical factors from [SS04, (6.2)] and [DD15, Theorems 1.2 and 8.2] rather than recomputing them. After Lemmas 2.2–2.4, the formula gives |Sel_φ̂| = 4·(Ω_E/Ω_E'), so the proof needs Ω_E/Ω_E' = 1/2. This ratio is model-dependent: for the non-minimal model (2.2), the pullback multiplier is c = 1, whereas for the minimal model introduced in Lemma 2.3 the multiplier is c = 1/2. If [DD15, Theorem 1.2] were applied to the wrong model, or if the |c|-dichotomy were not applicable to this 2-isogeny, the parity argument in Lemma 2.5 would be invalid; the resulting |Sel_φ̂| = 4 would allow |Sel_2| up to 4 and would not force rank 0. The paper states the minimal-model convention but does not reproduce the hypotheses of [DD15], and no numerical verification of the Selmer bounds is given. This is the only step where a normalization slip would break the central theorem while the surrounding 2-descent is visibly correct.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a nonisotrivial elliptic surface E/Q(T) defined by y^2 = x(x^2 - x + T) and proves Theorem 1.2: the group E(Q(T)) has rank 0, and there are infinitely many t in Q for which the specialized curve E_t has rank 0. The proof specializes at t = (m+n)/(2m), where m, m+n, m+2n are primes congruent to 3 mod 8; Green's theorem on 3-term arithmetic progressions in primes supplies infinitely many such triples. For these t, the curve E_t is isomorphic to the curve E: y^2 = x(x^2 - 4m^2 x + 8m^3(m+n)). The paper then performs a 2-descent on E, introducing a 2-isogenous curve E' and computing the associated Selmer groups. The local solvability computation (Lemma 2.4) shows |Sel_phi(E/Q)| = 2, and a Cassels period-ratio argument (Lemma 2.5) shows |Sel_phihat(E'/Q)| = 2. The exact sequence relating these Selmer groups to Sel_2(E/Q) then gives |Sel_2(E/Q)| <= 2, which forces the rank of E(Q) to be 0.","tokens_in":8701,"tokens_out":61829,"duration_ms":503578,"significance":"If correct, this is the first unconditional example of Conjecture 1.1 for a nonisotrivial elliptic curve over Q(T): infinitely many fibers have rank equal to the generic rank (here 0), rather than the generic rank being exceeded. The proof is largely self-contained: the 2-descent is spelled out in detail, the local root numbers and Tamagawa numbers are obtained via Tate's algorithm and cited tables, and the only external inputs are Green's theorem, Cassels' formula, and the Dokchitser-Dokchitser results. The construction is not tuned to the conclusion; the same descent works for every admissible triple. The result directly addresses a conjecture in the literature and is likely to be of significant interest.","major_comments":[{"comment":"Lemma 2.5 does not specify which Weierstrass model of E' is used to define the differential ω' and the period Ω_E'. The model (2.2) is not minimal, and if the period is taken from that model while the Tamagawa numbers are taken from the minimal model of Lemma 2.3, the Cassels formula is inconsistent. For the non-minimal model (2.2), the pullback satisfies φ*ω' = -ω, so the constant c defined by c·φ*ω' = ω would have |c| = 1, leading to |Sel_phihat| = 4, which would not force the rank bound. The proof should state explicitly that E' is taken with the minimal model of Lemma 2.3, that the isogeny is composed with the isomorphism to that model, and that for this model the pullback multiplier is -2 (so c = -1/2 and |c| = 1/2). With this clarification the period-ratio step is correct.","section":"§2, Lemma 2.5"}],"minor_comments":[{"comment":"In the sentence 'the Kodaira symbols of E at 2, m, m+n and m+2n are equal to II, III*, I1 and I2', the letter E should be E'.","section":"§2, Lemma 2.3"},{"comment":"The equation 'c · φ*ω' = ω' is easy to misread; it would be clearer to write 'φ*ω' = (1/c) ω' or to state explicitly that c is the reciprocal of the pullback multiplier.","section":"§2, Lemma 2.5"},{"comment":"The computation of W_2(E) relies on Halberstadt's Table 1 and a specific list of invariants, but the table is not reproduced. This is acceptable, but a brief indication of how the table is read would improve readability.","section":"§2, Lemma 2.2"},{"comment":"The remark that (1,b) is a point of infinite order on E_{b^2} for all but finitely many b is terse; one or two sentences explaining the use of Silverman's specialization theorem would be helpful.","section":"§1"},{"comment":"The paper does not give a numerical example (e.g., m=3,n=8) to illustrate the Selmer computation. Adding one would make the descent concrete and easier to follow.","section":"§2"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound and the result is significant. The main review concern was the model dependence in Lemma 2.5; on a careful reading, the intended use of the minimal model for E' is correct, and the period ratio Ω_E/Ω_E' = 1/2 follows from the pullback multiplier -2 and the fact that the two real components of E' have equal periods. However, the proof should clarify this explicitly, since a reader applying the formula to the non-minimal model (2.2) would obtain the wrong value. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should read this one. Zywina proves the first unconditional case of the rank-non-jumping conjecture: for E/Q(T) : y^2 = x(x^2 - x + T), the generic rank is 0 and E_t(Q) has rank 0 for infinitely many t in Q. Earlier examples were conditional on unproved prime conjectures (Mersenne primes, or primes t^2+64). The new idea is to choose t from Green's theorem on 3-term arithmetic progressions in primes, so that the specialized curve has only four bad primes, all congruent to 3 mod 8, and then run an explicit 2-descent.\n\nWhat is genuinely good: the descent is worked out rather than black-boxed. Lemmas 2.2-2.4 compute the root number, Tamagawa products, and the phi-Selmer group by hand using Tate's algorithm. Lemma 2.4 is a clean local argument that eliminates all square classes except ±1 and ±m(m+2n), and then gets rid of −1 by a valuation argument at p=m+n. The use of Green's theorem is exactly right: it gives infinitely many admissible triples because the primes lie in a fixed residue class.\n\nThe soft spot, as you might have guessed from the title, is Lemma 2.5. The bound |Sel_phihat(E'/Q)| = 2 comes from the Cassels-Schaefer-Stoll period formula together with the Dokchitser-Dokchitser theorem that the scaling factor |c| lies in {1, 1/2}. The paper states the minimal-model convention, but it does not show the transformation between the model used to define the isogeny and the minimal model of E'. So the reader has to trust that the |c| dichotomy applies to the right period ratio. I checked it: the minimal model for E' is obtained by (x', y') = (4X, 8Y), the differential on the original model is half the minimal differential, and the pullback scaling for the minimal models is |c| = 1/2, not 1. So Lemma 2.5 is correct, but a referee should ask for that one sentence to be spelled out. This is a minor clarity issue, not a load-bearing flaw.\n\nThe citation pattern is fine. [Zyw25] is an announced followup, not used. The earlier conditional results are properly credited.\n\nThis paper is for arithmetic geometers working on elliptic surfaces, Selmer groups, or rank jumps. It resolves a special case of Conjecture 1.1 and the method might generalize, as the announced rank-2 followup suggests. I would send it to a serious referee, with a specific request to verify the normalization in Lemma 2.5. It deserves acceptance after a small revision.\n\nBest","headline":"First unconditional example of rank-non-jumping fibers for a nonisotrivial elliptic surface; the 2-descent is solid and the only soft spot is a normalization detail in Lemma 2.5 that is correct but should be spelled out.","tokens_in":9195,"tokens_out":11787,"would_cite":true,"duration_ms":88389,"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":"The elliptic curve $y^2 = x(x^2 - x + T)$ over $\\mathbb{Q}(T)$ is proved to have rank 0, and infinitely many rational specializations also have rank 0, giving the first unconditional instance of Conjecture 1.1.","keywords":["elliptic curves","elliptic surfaces","rank of elliptic curves","specialization","Selmer group","2-descent","root number","prime arithmetic progressions"],"falsifier":"Compute the 2-Selmer group of the curve $E_t$ for $t = 11/6$, obtained from the prime triple $(3, 11, 19)$; the proof predicts $|\\mathrm{Sel}_2| = 2$ and rank 0. If a rational point of infinite order exists, or if the Selmer group has order divisible by 4, then Lemma 2.5's period-ratio application is wrong and Theorem 1.2 fails.","tokens_in":8205,"feed_emoji":"","tokens_out":14159,"duration_ms":118094,"temperature":0.7,"pith_summary":"The paper proves the first unconditional case of a conjecture about ranks of fibers in a nonisotrivial elliptic family: for $E/\\mathbb{Q}(T)$ defined by $y^2 = x(x^2 - x + T)$, the generic rank is 0 and infinitely many rational fibers $E_t(\\mathbb{Q})$ also have rank 0. This matters because it had been open whether any nonisotrivial family could have infinitely many fibers on which the rank does not jump; all earlier constructions required an unproved number-theoretic assumption. The proof takes $t = (m+n)/(2m)$ where $m$, $m+n$ and $m+2n$ are primes congruent to 3 mod 8, so that $E_t$ has good reduction away from $\\{2, m, m+n, m+2n\\}$, and then bounds the rank by a 2-descent. The infinitude of such prime triples follows from a theorem on 3-term arithmetic progressions in the primes, giving the infinitely many $t$.","feed_headline":"Rank 0 repeats infinitely often on one elliptic surface","feed_subtitle":"It is the first unconditional proof that a nonisotrivial elliptic surface has infinitely many fibers of matching rank.","key_machinery":"The argument is carried by a 2-descent through the cyclic 2-isogeny $\\phi: E \\to E'$, whose kernel is generated by the 2-torsion point $(0,0)$. The $\\phi$-Selmer group is computed directly from the homogeneous spaces $C_d$ and their local solubility, and has order 2. The dual Selmer group $\\mathrm{Sel}_{\\hat{\\phi}}(E'/\\mathbb{Q})$ is controlled by the Schaefer-Stoll formula, which expresses $|\\mathrm{Sel}_{\\hat{\\phi}}|/|\\mathrm{Sel}_{\\phi}|$ as the product of the period ratio $\\Omega_E/\\Omega_{E'}$, a Tamagawa-number ratio, and a torsion ratio; the period ratio is forced to be $1/2$ by a theorem on local invariants of isogenous elliptic curves, because the alternative 1 would contradict the computed root number $W(E) = 1$. An exact sequence then bounds $\\mathrm{Sel}_2(E/\\mathbb{Q})$ by $\\mathrm{Sel}_{\\hat{\\phi}}(E'/\\mathbb{Q})$, giving $|\\mathrm{Sel}_2| \\leq 2$. The infinite supply of suitable parameters $t$ comes from a theorem on 3-term arithmetic progressions in the primes congruent to 3 mod 8.","core_discovery":"The central claim, Theorem 1.2, is that the elliptic curve $E$ over $\\mathbb{Q}(T)$ given by $y^2 = x(x^2 - x + T)$ has $E(\\mathbb{Q}(T))$ of rank 0, and that $E_t(\\mathbb{Q})$ has rank 0 for infinitely many $t \\in \\mathbb{Q}$. For any prime triple $(m, m+n, m+2n)$ all congruent to 3 mod 8, the paper sets $t = (m+n)/(2m)$ and shows that the specialized curve is isomorphic to $y^2 = x(x^2 - 4m^2x + 8m^3(m+n))$. The rank bound is obtained by computing the 2-Selmer group through the cyclic 2-isogeny $\\phi: E \\to E'$, using the Schaefer-Stoll formula and a period-ratio theorem for isogenous elliptic curves to obtain $|\\mathrm{Sel}_2(E/\\mathbb{Q})| \\leq 2$; since $E(\\mathbb{Q})$ has a point of order 2, the rank must be 0. The infinitude of the prime triples, given by Green's theorem on 3-term arithmetic progressions in primes, makes the set of such $t$ infinite.","pith_inferences":["One could look for other residue classes $a \\bmod 8$ and other signature choices where the relevant Legendre symbols give the same period-ratio and root-number configuration; this would produce further unconditional families satisfying Conjecture 1.1.","The proof uses the period ratio only through its possible values $\\{1, 1/2\\}$; a natural numerical experiment is to compute $\\Omega_E/\\Omega_{E'}$ for the first few prime triples to see the ratio is always $1/2$, which would test the single most delicate step of the descent.","A quantitative version of the argument might supply an explicit positive-density statement for the rank-0 fibers in this family, since the prime triples from Roth-type theorems are numerous; the paper itself only needs infinitude."],"forward_implications":["Conjecture 1.1 holds for the specific surface $E$, giving the first unconditional confirmation of that conjecture.","For every prime triple $m, m+n, m+2n \\equiv 3 \\pmod{8}$, the fiber $E_t$ at $t = (m+n)/(2m)$ has 2-Selmer group of order exactly 2 and rank exactly 0.","Silverman's specialization bound, together with infinitely many rank-0 fibers, determines the generic rank of $E(\\mathbb{Q}(T))$ to be 0.","The method of confining bad reduction to a few explicit primes, then running a 2-descent, is carried over to a rank-2 example in a follow-up paper, so the approach is not restricted to rank 0."],"supporting_citations":[{"why":"Supplies the infinitude of 3-term arithmetic progressions in the primes congruent to 3 mod 8, which generates the infinitely many $t$.","marker":"[Gre05]"},{"why":"Gives the period ratio $\\Omega_E/\\Omega_{E'} \\in \\{1,1/2\\}$ and the parity statement used in Lemma 2.5 to force the ratio $1/2$.","marker":"[DD15]"},{"why":"Provides the formula for $|\\mathrm{Sel}_{\\hat{\\phi}}|/|\\mathrm{Sel}_{\\phi}|$ in terms of periods and Tamagawa numbers, and the exact sequence relating $\\mathrm{Sel}_{\\phi}$, $\\mathrm{Sel}_2$, and $\\mathrm{Sel}_{\\hat{\\phi}}$.","marker":"[SS04]"},{"why":"Gives the specialization bound that the generic rank is at most the rank of all but finitely many fibers, used to conclude $r = 0$.","marker":"[Sil83]"},{"why":"Supplies the local root number formulas used to compute $W(E) = 1$ in Lemma 2.2.","marker":"[Roh93]"},{"why":"Provides the table of local root numbers at 2 and 3 used for the primes 2 and possibly 3 in Lemma 2.2.","marker":"[Hal98]"},{"why":"The earlier conditional example whose Mersenne-prime assumption the present result removes; the baseline for 'first unconditional'.","marker":"[CP23]"}],"fun_headline_variants":["First proof of infinitely many rank-zero fibers on an elliptic surface","Rank-zero fibers infinitely often via prime arithmetic progressions","Infinite rank-zero fibers via Green's prime progressions","First unconditional proof of infinite rank-zero fibers","Prime triples give infinite rank-zero specializations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rank-zero conclusion rests on the period-ratio theorem for isogenous elliptic curves being exactly applicable, so that $\\Omega_E/\\Omega_{E'} = 1/2$ and never 1 for any constructed curve; if the ratio ever equaled 1, the Selmer bound $|\\mathrm{Sel}_{\\hat{\\phi}}(E'/\\mathbb{Q})| = 2$ would fail and the rank could be positive.","fun_headline_variants_meta":{"raw":{"variants":["First proof of infinitely many rank-zero fibers on an elliptic surface","Rank-zero fibers infinitely often via prime arithmetic progressions","Infinite rank-zero fibers via Green's prime progressions","First unconditional proof of infinite rank-zero fibers","Prime triples give infinite rank-zero specializations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00094,"raw_usage":{"total_tokens":4050,"prompt_tokens":1009,"completion_tokens":3041,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":2965}},"tokens_in":625,"tokens_out":3041,"duration_ms":23662,"temperature":1.0,"reasoning_tokens":2965,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T16:51:51.782774+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the 2-Selmer group of the curve $E_t$ for $t = 11/6$, obtained from the prime triple $(3, 11, 19)$; the proof predicts $|\\mathrm{Sel}_2| = 2$ and rank 0. If a rational point of infinite order exists, or if the Selmer group has order divisible by 4, then Lemma 2.5's period-ratio application is wrong and Theorem 1.2 fails.","supporting_citations":[],"review_version":1}