{"id":"5d230e4a-f079-4da2-aeb5-2fdaf111777f","arxiv_id":"2505.16960","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"There are infinitely many elliptic curves over any number field with rank 1, and for any quadratic extension L/K, infinitely many curves satisfy rank E(K) = rank E(L) = 1.","lead":"For every quadratic extension L/K of number fields, this paper proves there are infinitely many elliptic curves E over K such that E(K) and E(L) both have rank 1. The proof specializes a fixed rank-1 family and uses explicit 2-descents, giving an unconditional construction of infinitely many rank-1 elliptic curves over any number field.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem hinges on an unverified black-box application of [Kai25, Prop. 13.2] in Lemma 4.1; the admissibility hypotheses and the real-place formulation of the local conditions are not checked, so acceptance should be conditional on that check.","rationale":"The central construction is a coherent 2-descent: Theorem 1.5 reduces the rank statements to local conditions; Section 3 explicitly constructs S and U_p; Lemmas 3.9-3.11 compute Selmer ratios and generators; the rank conclusion from the exact sequence (2.2) and equations (3.17)-(3.18) is sound. I did not find an internal inconsistency or a post-hoc fitting. The weakest link is the black-box citation in Lemma 4.1: the proof does not show that the local sets U_p satisfy the hypotheses of [Kai25, Prop. 13.2], and the real-place conditions are asserted without specifying how they fit into Kai's finite-place framework. This is exactly the reader's weakest assumption, so I agree with the reader's identification. Because this is a load-bearing dependence on an external preprint, I would phrase the recommendation as CONDITIONAL rather than unconditional ACCEPT: verify Prop. 13.2's hypotheses for these U_p and, as a sanity check, the Tate's algorithm cases in Lemma 2.2. The paper is transparent and the computations are explicit, so this is a verification condition, not a rejection.","tokens_in":20671,"tokens_out":32623,"duration_ms":265705,"concrete_test":"Read [Kai25, Prop. 13.2] and its definitions; then, for the sets S and U_p of Section 3, verify in writing: (1) each U_p is nonempty open in K_p^2 and the real conditions |a/b - 1/2|_v < epsilon, b > 0 define a nonempty open subset of K_v^2; (2) the proposition's admissibility and local-density hypotheses hold for the forms X, Y, X+Y, X-Y; (3) the conclusion yields a,b in O_{K,S} with a, a+b, a-b generating distinct nonzero prime ideals of O_{K,S}, not merely prime elements. If any hypothesis fails, adjust or replace Lemma 4.1; if all checks pass, the concern is resolved and the reader's ACCEPT stands.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Lemma 4.1 selects (a_p,b_p) in U_p and then asserts that [Kai25, Prop. 13.2] produces a,b in O_{K,S} with: closeness to (a_p,b_p) at every p in S, |a/b - 1/2|_v < epsilon and b > 0 at every real place, and a, a+b, a-b generating distinct prime ideals of O_{K,S}. The paper never states the hypotheses of Prop. 13.2, never verifies that the U_p built in Section 3 are admissible for those hypotheses, and does not specify how the real-place conditions are encoded as a finite set of places for Kai's theorem. Since Lemma 4.1 supplies the only input to Theorem 1.5, and Theorem 1.5 is the engine for Theorems 1.1 and 1.2, a failure or inapplicability of Prop. 13.2 would leave Theorems 1.1 and 1.2 without proof. This is a verification gap rather than a demonstrated contradiction; however, it is the single point on which the central claim is least secure. A secondary but related gap is that the nine Tate's algorithm cases in Lemma 2.2 are summarized rather than displayed; an error in any c_p ratio would also break the Selmer computations, although the models given make those cases checkable.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for any quadratic extension L/K of number fields there are infinitely many elliptic curves E over K, up to isomorphism, with rank E(K) = rank E(L) = 1 (Theorem 1.2), and in particular infinitely many rank-one elliptic curves over any number field (Theorem 1.1). The proof constructs a nonisotrivial family of elliptic curves depending on parameters a,b and uses an explicit 2-descent to show that, under suitable local conditions collected in Theorem 1.5, the curve E1 has rank 1 while its quadratic twist ED by D has rank 0. The local conditions are then realized by applying a result of Kai on linear patterns of prime elements in number fields, giving the required a,b. The paper also explains how the rank stability statement implies rank-one curves and how shrinking the local conditions yields infinitely many non-isomorphic examples.","tokens_in":20925,"tokens_out":30576,"duration_ms":251876,"significance":"If the proof is complete, this is a substantial result: it removes the restrictions in Koymans–Pagano (many real places) and gives a curve-based analogue of the Alpöge–Bhargava–Ho–Shnidman rank stability theorem, with the same applications to Hilbert's tenth problem over rings of integers. The approach is genuinely different: instead of averaging over families of twists, it specializes a nonisotrivial rank-one family and computes the relevant Selmer groups explicitly. A notable strength is that the descent argument is explicit and checkable, and the main theorems do not rely on unproved conjectures such as finiteness of the 2-part of the Tate–Shafarevich group. The proof is essentially self-contained apart from standard tools and the cited theorem of Kai; the author's earlier papers [Zyw25a], [Zyw25b] are used for framework and ideas, not as a circular dependency.","major_comments":[{"comment":"The proof of Lemma 4.1 is a direct invocation of [Kai25, Proposition 13.2], but the manuscript neither states the hypotheses of that proposition nor verifies them for the sets U_p constructed in Section 3. In particular, one needs to check that the U_p are admissible for Kai's theorem; that the valuation and square conditions such as (v_p(a),v_p(a+b),v_p(a-b))=(-1,1,-1) and -2a(a+b) square are compatible with finding a,b in O_{K,S}; that the real-place inequalities |a/b-1/2|_v<epsilon and b>0 are encoded in the allowed archimedean local data; and that the conclusion 'a, a+b and a-b generate distinct prime ideals of O_{K,S}' is exactly the form of prime-element output supplied by the proposition. Since Lemma 4.1 is the only bridge from the purely local Theorem 1.5 to the global existence statements of Theorems 1.1 and 1.2, this is a load-bearing point. Please state Proposition 13.2 and verify its hypotheses explicitly.","section":"Section 4, Lemma 4.1"},{"comment":"The nine local cases of Lemma 2.2 are resolved by Tate's algorithm with the relevant Weierstrass models and Kodaira symbols stated but not derived. The split-versus-nonsplit multiplicative reduction determinations in cases (iv)-(ix) are especially important because they control the sizes of Im(delta_{d,p}) that feed into the identities |Phi_{1,p}|/|Phi_{D,p}| = xi_p in Lemma 3.1 and into the Selmer ratio computation in Lemma 3.9. I would like the Tate algorithm steps displayed, or at least a table giving the resulting coefficients and split/nonsplit criterion for each of these cases, so that the c_p ratios are directly checkable without reproducing the computation.","section":"Section 2.2, Lemma 2.2"}],"minor_comments":[{"comment":"The text says 'we take any a,b in O_K' but Theorem 1.5 and the subsequent arguments with ideals of O_{K,S} require a,b to be in O_{K,S}; this should be corrected.","section":"Section 3.3"},{"comment":"After Lemma 3.1, the set S1 has been enlarged to S1 union S'_1 during the proof of part (g), but the notation S1 is subsequently used without explicitly saying that it now denotes the enlarged set; please make this replacement explicit.","section":"Section 3.2"},{"comment":"There is a duplicated phrase 'as in in §2.2' in the paragraph introducing delta_{d,v}; this is a typo.","section":"Section 2.2"}],"recommendation":"major_revision","confidential_remarks":"The central descent argument appears sound and the paper is well within the scope of the journal. The only reason I am not recommending acceptance as-is is the unverified black-box use of [Kai25, Proposition 13.2] in Lemma 4.1. If the author states the proposition's hypotheses and checks them for the U_p constructed in Section 3, I would expect the paper to be acceptable. The compressed Tate algorithm cases in Lemma 2.2 are secondary but should also be expanded for referee and reader verification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a notable unconditional result — infinitely many rank-1 elliptic curves over any number field, and a stronger rank-stability statement over every quadratic extension. It deserves a serious referee and, after one specific check, publication.\n\nWhat's new: The main theorem generalizes the Koymans–Pagano and Alpöge–Bhargava–Ho–Shnidman results, and the method is genuinely different: instead of quadratic twists of a fixed curve, Zywina specializes a nonisotrivial rank-1 elliptic surface and does fully explicit 2-descents. The construction in Section 3 is carefully engineered to make the local Selmer computations come out; Lemma 3.10 and 3.11 give clean determinations of the relevant Selmer groups. The paper is transparent about prior work, including the author's own [Zyw25a,b], and the self-citations don't carry the argument.\n\nThe soft spots are both at the level of verification, not logic. The first is Lemma 4.1. The paper's existence of a,b in O_{K,S} with all the required local conditions is taken from Kai's Proposition 13.2, but the hypotheses of that proposition are never stated, and it's not verified that the open sets U_p constructed in Section 3 are admissible for it. This is a real gap in exposition. The stress-test note is right that this is the load-bearing point: if the proposition doesn't apply, Theorems 1.1 and 1.2 lose their engine. I don't think the paper is wrong — Kai's result is designed for precisely this kind of prime-ideal selection and the U_p are defined by congruence conditions — but the referee should ask for a spelled-out verification. It might be a short addition; it shouldn't be left to the reader.\n\nThe second is Lemma 2.2: nine cases of Tate's algorithm are summarized with Kodaira symbols but not walked through. This is a minor issue because the relevant Weierstrass models are written down, and the ratios of Tamagawa numbers are the only inputs. It's checkable, and the stated results look right.\n\nThe rank formula for the quadratic twist is standard, and the application to Hilbert's tenth problem is properly credited to the earlier work. The paper is honest and doesn't overclaim.\n\nWho it's for: number theorists working on Selmer groups, ranks of elliptic curves, and applications to Hilbert's tenth problem. I'd certainly cite it once the Kai check is settled.\n\nRecommendation: send to peer review; ask for a revision that explicitly verifies the hypotheses of Kai's Proposition 13.2 in Lemma 4.1.","headline":"A strong, genuinely new unconditional rank-1 existence theorem; the only substantive concern is the unverified hypotheses in the black-box use of Kai's proposition, which a referee should check.","tokens_in":21494,"tokens_out":2609,"would_cite":true,"duration_ms":20567,"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":"For every number field K and every quadratic extension L/K, this paper constructs infinitely many elliptic curves with rank exactly 1 over both fields.","keywords":["elliptic curves","rank one","quadratic twists","rank stability","2-descent","Selmer groups","number fields","Hilbert's tenth problem"],"falsifier":"Carry out the Section 3 construction for a concrete quadratic extension such as $\\mathbb{Q}(i)/\\mathbb{Q}$, then search systematically for $a,b\\in\\mathcal{O}_{K,S}$ satisfying the three conditions of Theorem 1.5; if some data set $(S,\\{U_p\\})$ admits no such pair, Lemma 4.1 fails and with it both main theorems. A more direct test is to verify the admissibility hypotheses of the cited Proposition 13.2 for the particular $U_p$ produced in Section 3; any instance where those hypotheses fail would expose a fatal gap in the proof.","tokens_in":20427,"feed_emoji":"🧮","tokens_out":14902,"duration_ms":107425,"temperature":0.7,"pith_summary":"The paper proves that for any quadratic extension $L/K$ of number fields there are infinitely many elliptic curves $E$ over $K$, up to isomorphism, with $\\mathrm{rank}\\,E(K)=\\mathrm{rank}\\,E(L)=1$. This generalizes earlier theorems that needed restrictions such as many real places or that worked with general abelian varieties, and it gives the smallest possible positive rank. The proof specializes a fixed elliptic surface over $\\mathbb{Q}(T)$ at infinitely many $S$-integral parameters $t=a/b$, and uses a fully computed 2-descent to show that the specialized curve has rank 1 while its quadratic twist by $D$ (with $L=K(\\sqrt{D})$) has rank 0. Because rank in a quadratic extension splits as the rank over $K$ plus the rank of the twist, this gives rank 1 over $L$, and shrinking the allowed parameter sets yields infinitely many distinct $j$-invariants.","feed_headline":"Infinitely many curves keep rank 1 under any quadratic extension","feed_subtitle":"A 2-descent construction over any number field yields infinitely many elliptic curves whose rank stays 1 in a quadratic extension.","key_machinery":"The carrying object is the degree-2 isogeny $\\phi_d: E_d \\to E'_d$ attached to the Weierstrass model (3.1), together with the Selmer groups $Sel_{\\phi_d}(E_d/K)$ and $Sel_{\\hat\\phi_d}(E'_d/K)$. The paper computes the ratio $|Sel_{\\phi}|/|Sel_{\\hat\\phi}| = \\prod_v \\frac{1}{2}|\\mathrm{Im}\\,\\delta_v|$ from local images $\\delta_v$ that are pinned down by Tate's algorithm (Lemma 2.2) and by the product formula of Lemma 2.1. The auxiliary finite sets $S_0,S_1,S_2,S_3$ and open sets $U_p$ are chosen to force these local images so that the Selmer elements reduce to $\\{1, 2a(a+b)\\}$ for both $d=1$ and $d=D$, and so that the exact sequence (2.2) gives $E_1(K)/2E_1(K)\\cong(\\mathbb{Z}/2\\mathbb{Z})^2$ and $E_D(K)/2E_D(K)\\cong\\mathbb{Z}/2\\mathbb{Z}$, forcing ranks $1$ and $0$; the point $P_t=(-2t(t+1), 2t(t+1)^2)$ provides the rational point of infinite order via the specialization theorem.","core_discovery":"The central claim is that rank stability with the smallest possible positive rank is universal: over any quadratic extension $L/K$ of number fields, infinitely many elliptic curves $E$ up to isomorphism satisfy $\\mathrm{rank}\\,E(K)=\\mathrm{rank}\\,E(L)=1$. The curves are explicit specializations of the surface $y^2=x^3+4T(T+1)x^2+2T(T+1)^2(T-1)x$, specialized at parameters $t=a/b$ where $a,b\\in\\mathcal{O}_{K,S}$ and $a$, $a+b$, $a-b$ generate distinct prime ideals. For such parameters an explicit 2-descent through the 2-isogeny shows that the curve $E_1$ has rank exactly 1 over $K$ and its quadratic twist $E_D$ by the element $D$ with $L=K(\\sqrt{D})$ has rank 0. Since $\\mathrm{rank}\\,E_1(L)=\\mathrm{rank}\\,E_1(K)+\\mathrm{rank}\\,E_D(K)$, the curve has rank 1 over $L$; shrinking the allowed open sets $U_p$ one $j$-invariant at a time produces infinitely many non-isomorphic examples.","pith_inferences":["Going beyond the paper: replacing the cited number-field additive-combinatorial existence result by an effective sieve could turn the theorem into a quantitative counting statement for rank-1 curves of bounded height; the local setup seems designed for such a refinement.","Going beyond the paper: the proof establishes rank stability for one quadratic twist at a time, and it is natural to ask whether the same Selmer-ratio bookkeeping can force several twists $E_{D_1},E_{D_2},\\dots$ to have rank 0 simultaneously, yielding curves that keep rank 1 under multiple quadratic extensions.","Going beyond the paper: the role of the three-term progression $a-b,a,a+b$ suggests that longer additive patterns, if produced by the same number-field machinery, would give rank-stable families with more than one free parameter."],"forward_implications":["For any number field $K$, there are infinitely many elliptic curves of rank exactly 1, unconditionally and without conjectures on Tate–Shafarevich groups.","For any quadratic extension $L/K$, infinitely many curves satisfy $\\mathrm{rank}\\,E(K)=\\mathrm{rank}\\,E(L)=1$, the smallest positive rank at which rank stability can occur.","The curves are explicit specializations of one fixed Weierstrass family, so the 2-descent computes the ranks exactly rather than merely bounding them.","The same rank-stability input that earlier theorems used for Hilbert's tenth problem over rings of integers is now obtained uniformly for every number field.","Infinitely many non-isomorphic examples follow by excluding finitely many $j$-invariants from the parameter sets, so the count is genuinely infinite up to isomorphism."],"supporting_citations":[{"why":"Supplies Proposition 13.2, the number-field additive-combinatorial statement that produces the required $a,b \\in O_{K,S}$ with prescribed local approximations and distinct prime ideals $a$, $a\\pm b$.","marker":"[Kai25]"},{"why":"Height/specialization theorem proving the point $P_t$ has infinite order on all but finitely many fibers, giving the lower bound $\\mathrm{rank}\\,E(K)\\ge 1$.","marker":"[Sil83]"},{"why":"Background for 2-descent via a 2-isogeny: the Selmer group definition, the connecting homomorphism $\\delta$, and exact sequence (2.2).","marker":"[Sil09]"},{"why":"Product formula equating $|Sel_\\phi|/|Sel_{\\hat\\phi}|$ with a product of local terms, the global Selmer-ratio tool used throughout Section 3.","marker":"[Cas65]"},{"why":"Local invariants of isogenous curves used in Lemma 2.1 to identify $\\frac12|\\mathrm{Im}\\,\\delta_p|$ with $c_p(E')/c_p(E)$ and to treat real places.","marker":"[DD15]"},{"why":"Tate's algorithm as applied in Lemma 2.2 to compute Tamagawa numbers and Kodaira types at each relevant prime.","marker":"[Sil94]"},{"why":"Reciprocity law and unit-group structure used in Lemmas 3.3 and 3.7 for the parity and nonsquare conditions involving $\\pi_1$ and $q_3$.","marker":"[Neu99]"},{"why":"Earlier rank-2 construction over $\\mathbb{Q}$ whose techniques (specializing a nonisotrivial family and controlling Selmer growth) are adapted here to rank 1 over number fields.","marker":"[Zyw25b]"}],"fun_headline_variants":["Rank-1 curves stack up: infinite family stable under quadratic extensions","Infinite rank-1 elliptic curves hold rank across every quadratic extension","Rank-1 family over every number field, invariant under quadratic twists","Quadratic extension? Rank stays 1: infinite such curves over any number field"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof hinges on the referenced proposition that guarantees, for the specific finite set $S$ and open sets $U_p$ constructed in Section 3, the existence of $S$-integers $a,b$ with prescribed local approximations and with $a$, $a+b$, $a-b$ generating distinct prime ideals; the paper cites that proposition without checking all of its admissibility hypotheses for these $U_p$, so if the proposition is false or inapplicable, Lemma 4.1 and both main theorems collapse.","fun_headline_variants_meta":{"raw":{"variants":["Rank-1 curves stack up: infinite family stable under quadratic extensions","Infinite rank-1 elliptic curves hold rank across every quadratic extension","Rank-1 family over every number field, invariant under quadratic twists","Quadratic extension? Rank stays 1: infinite such curves over any number field"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000644,"raw_usage":{"total_tokens":2947,"prompt_tokens":918,"completion_tokens":2029,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":1951}},"tokens_in":534,"tokens_out":2029,"duration_ms":12267,"temperature":1.0,"reasoning_tokens":1951,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:53:10.210495+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Carry out the Section 3 construction for a concrete quadratic extension such as $\\mathbb{Q}(i)/\\mathbb{Q}$, then search systematically for $a,b\\in\\mathcal{O}_{K,S}$ satisfying the three conditions of Theorem 1.5; if some data set $(S,\\{U_p\\})$ admits no such pair, Lemma 4.1 fails and with it both main theorems. A more direct test is to verify the admissibility hypotheses of the cited Proposition 13.2 for the particular $U_p$ produced in Section 3; any instance where those hypotheses fail would expose a fatal gap in the proof.","supporting_citations":[],"review_version":1}