{"id":"cef20429-6d8c-4390-8d79-193e1a64102d","arxiv_id":"2504.21799","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under Iwasawa-main-conjecture hypotheses, a rank-one elliptic curve over a real quadratic field with finite Tate-Shafarevich group has analytic rank one.","lead":"The paper proves that, for certain elliptic curves over real quadratic fields, having exactly one independent rational point and a finite Tate-Shafarevich group forces the L-function to have a simple zero at its center. This establishes a p-converse to the Gross-Zagier-Kolyvagin theorem in a setting beyond the rational numbers.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorem rests on the unproven minimal modular lifting hypothesis (MML); the only verification in Section 11 is a base change from Q, so the genuinely new real-quadratic case remains conditional on this extra input.","rationale":"The reader identified (MML) as the weakest assumption, and my stress-test agrees. I checked the main proof chain: the choice of K in Lemma 8.1, the control theorem Proposition 4.1, Corollary 5.2 on the order of vanishing of L^cc_p, the p-adic weight pairing computations in Section 7, and the final length inequalities in Lemma 9.1 and Theorem 9.2. I did not find an internal contradiction or a missing argument that would move the verdict to REJECT or UNVERDICTED. The paper is explicit that (MML) is a hypothesis in Theorem 1.3. However, because (MML) is needed at the crucial step where Wan's main conjecture supplies the inequality (1), and because the only example verifying (MML) is a base change from Q, the theorem's reach beyond base-change curves is conditional on an unproven modular lifting statement. This matches the reader's CONDITIONAL verdict at moderate confidence, so I recommend no change to the verdict.","tokens_in":34386,"tokens_out":35905,"duration_ms":406507,"concrete_test":"Check whether (MML) is unconditionally available for GL_2 over a real quadratic F in this setting (p>5 inert, split multiplicative reduction, (irred)) via existing Hilbert modularity lifting theorems or minimal R=T results. If yes, supply the proof or precise citation and Theorem 9.2 becomes unconditional. If not, construct or identify a non-base-change example where (MML) can be verified, or explicitly state in the paper that the main theorem is conditional on an open modular lifting problem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central implication rank_E(F)=1 plus finite III_p∞ => ord_{s=1} L(E/F,s)=1 depends on Corollary 4.2, which converts Wan's Iwasawa main conjecture into the inequality ord_{k=2} L^cc_p <= len_P(X^{S,cc}_{F∞}). Wan's Theorem 3.1, invoked here, requires (MML): the existence of a minimal modular lifting of the residual representation ρ̄_f (Section 3.3, Theorem 3.1(3)). This hypothesis is not proved in the paper for a general real quadratic field F. The only evidence offered is Section 11, where E is the base change of X0(11) from Q and a minimal modular lifting over Q supplies the required lifting. For a genuinely non-base-change E/F, no argument or citation is given that such a minimal modular lifting exists. Thus the theorem is internally consistent but its applicability to the advertised new setting is tied to an unverified modular lifting input. If (MML) fails, the inequality from Corollary 4.2 collapses and the proof of Theorem 9.2 cannot proceed; the paper's abstract and Theorem 1.3 do state (MML) as an assumption, so this is a limitation on scope rather than an internal contradiction, but it is the least secure load-bearing condition in the chain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a rank-one p-converse for elliptic curves over real quadratic fields, under two global hypotheses called (irred) and (MML). Assuming E/F has split multiplicative reduction at an inert prime p>5, rank one, and finite Tate-Shafarevich group, the authors show ord_{s=1} L(E/F,s)=1. The proof combines Wan's Iwasawa main conjecture for Hilbert modular forms, Mok's p-adic L-function and Heegner point results, and Nekovár's Selmer complexes. The authors also deduce a p-converse over Q and give a worked example with F=Q(√2), p=11, E=X0(11).","tokens_in":34703,"tokens_out":31802,"duration_ms":373598,"significance":"If the hypotheses are met, the result is a genuine extension of the Venerucci/Skinner-Zhang circle of ideas from Q to a real quadratic base field, and the chain of inequalities in §§4–9 is coherent and uses deep published results rather than fitted parameters. The paper's main contribution is the exact computation of a Selmer-complex length (Theorem 7.1), which combined with Wan's divisibility forces the order of vanishing to be exactly 4. The significance is tempered by two limitations: the theorem is explicitly conditional on the unproved minimal modular lifting hypothesis (MML), and the only numerical example verifying (MML) is a base change from Q, so the genuinely new real-quadratic case is not yet demonstrated. I find no internal contradiction in the main chain, but one load-bearing step in the application over Q needs additional justification.","major_comments":[{"comment":"The deduction that the hypotheses of Theorem 1.3 hold for the base-changed curve E/F is incomplete. Theorem 1.3 requires (irred) for E/F, but the proof only assumes (irred) for E/Q. Absolute irreducibility of a mod p representation does not automatically survive restriction to a quadratic field: a dihedral representation induced from a quadratic field becomes reducible over that field. The proof should either prove that ρ̄_{E,F} is absolutely irreducible for the particular F constructed, or explicitly choose F so as to avoid the finitely many quadratic fields over which the residual representation becomes reducible. This is load-bearing because Proposition 4.1 uses (irred) to obtain the vanishing of an H^0 that is essential for the control theorem.","section":"Section 10, proof of Theorem 10.1"},{"comment":"The main theorem is conditional on (MML), the existence of a minimal modular lifting of ρ̄_f, and this hypothesis is not proved in any genuinely new setting. The example in Section 11 is the base change of X0(11) from Q, so the required minimal lifting is inherited from Q. Since (MML) is exactly the input that supplies Wan's divisibility char ⊂ (L) and hence the inequality in Corollary 4.2, the advertised 'beyond Q' theorem remains conditional on an unverified modular lifting input for non-base-change real quadratic curves. This is not an internal contradiction, but the limitation should be stated prominently in the introduction and abstract, and the paper would be substantially strengthened by an unconditional instance of (MML) for a curve that does not arise as a base change.","section":"Section 3.3, Theorem 3.1(3), and Section 11"},{"comment":"The proof of Theorem 9.2 uses the assertion that p splits in K, which is equivalent to Ǩ(p)=1. This does follow from the conditions in Lemma 8.1 because nE=np, so the set of primes l dividing nED_F includes p and the condition ξ'(l)=1 for all such l forces Ǩ(p)=1. The manuscript should say this explicitly; as written, the reader is left to infer that the notation nE includes the inert prime p, and a misreading here would invalidate both Corollary 5.2 and the hypotheses of Theorem 7.1.","section":"Section 8, Remark 8.2(c), and Section 9"}],"minor_comments":[{"comment":"The Tate-Shafarevich group is typeset as 'ΠΠ' throughout the abstract and introduction; this should be corrected to a standard symbol such as III.","section":"Abstract and Section 1"},{"comment":"The step invoking [SU14, Corollary 3.8] to pass characteristic ideals modulo the ideal pcc is very terse; a short explanation of why the reduction is compatible for these Selmer groups would improve readability.","section":"Corollary 4.2"},{"comment":"The example relies on LMFDB data for the rank and the finiteness of the Tate-Shafarevich group and for the 'maximal image' assertion. Please indicate whether these are rigorous proven computations or numerical/conjectural data, since the example is meant to certify all hypotheses.","section":"Section 11"},{"comment":"In the proof of Theorem 10.1, the notation NE=Np makes the condition ξ(q)=1 for q|NE include q=p; given that the later argument requires ξ(p)=1 and ξ'(p)=-1, a parenthetical clarification would prevent confusion.","section":"Section 10"}],"recommendation":"major_revision","confidential_remarks":"The central conditional argument appears sound, and the authors clearly know the relevant literature. The main points to resolve in revision are the residual irreducibility of the base-changed representation in Section 10 and a transparent discussion of the scope of (MML). I do not see a fatal error, but the paper's novelty claim depends on these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is the first p-converse theorem for rank-one elliptic curves over a real quadratic field with split multiplicative reduction at an inert prime, and within its stated hypotheses the argument appears coherent. The authors extend Venerucci's strategy for Q using the four-variable p-adic L-function, Wan's Iwasawa main conjecture for Hilbert modular forms, Mok's Heegner-point formulas, and an extension of the p-adic weight pairing. The inequality chain is tidy: order of vanishing bounded below by 4, bounded above by a Selmer-complex length, and Theorem 7.1 computes that length as 2. The corollary over Q, Theorem 10.1, genuinely removes a hypothesis from Venerucci's theorem and looks convincing, since the minimal modular lifting there is inherited from Q.\n\nThe main soft spot is exactly the one the stress-test note flags: the proof of Corollary 4.2 uses Wan's main conjecture, which requires the minimal modular lifting hypothesis (MML). For a general real quadratic F and an E/F that is not a base change from Q, the paper gives no argument or reference that a minimal modular lifting exists. The authors are honest that (MML) is an assumption, so the theorem is conditional rather than circular, but the advertised 'beyond Q' content is therefore conditional on an unverified input. Section 11 verifies (MML) only for the base change of X0(11), which is not evidence for the genuinely new case. A referee should press on whether (MML) can be proved, or at least reduced to something known, for non-base-change curves.\n\nThere are minor soft spots too: a few checks are asserted rather than shown, such as the well-definedness of i-dagger-E, and the example relies on LMFDB data rather than a proof. These are minor, not load-bearing. The published theorems cited (Wan, Mok, Nekovar, Loeffler-Zerbes, Bergdall-Hansen) are appropriate, and I see no free parameters or circularity.\n\nWho is this for? Anyone working on p-converse theorems, Iwasawa main conjectures, or Heegner points over totally real fields. The paper deserves a serious referee. I would send it to peer review, not desk reject, with instructions to focus on the status and scope of (MML).","headline":"First p-converse over a real quadratic field in the multiplicative case, built on a coherent chain of deep published results, but the genuinely new non-base-change case rests on an unproved minimal modular lifting hypothesis.","tokens_in":35203,"tokens_out":1995,"would_cite":true,"duration_ms":26159,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G40","11G05","11R23"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for an elliptic curve over a real quadratic field with split multiplicative reduction at an inert prime p>5, rank-one Mordell–Weil group plus finite p-primary Tate–Shafarevich group forces the Hasse–Weil L-function…","keywords":["elliptic curves","real quadratic fields","p-converse theorem","p-adic L-functions","Iwasawa main conjecture","Selmer complexes","Tate-Shafarevich group","Hilbert modular forms"],"falsifier":"Take the paper's explicit example, the base change of the elliptic curve X0(11) to Q(\\sqrt{2}) with p=11, and compute the Hasse-Weil L-function near s=1; the theorem predicts L(E/F,1)=0 with a nonzero first derivative, so a computed order of vanishing different from 1 would refute the theorem. Alternatively, compute the order of vanishing at k=2 of the central critical p-adic L-function L_cc_p(f/K,k) for the auxiliary K, which the proof forces to be exactly 4.","tokens_in":88,"feed_emoji":"🔢","tokens_out":9557,"duration_ms":224715,"temperature":0.7,"pith_summary":"This paper aims to prove a p-converse theorem for elliptic curves over real quadratic fields: if the Mordell–Weil group has rank one and the p-primary Tate–Shafarevich group is finite, then the complex L-function vanishes to order one at s=1. The proof works when p>5 is inert in the field, the curve has split multiplicative reduction above p, and two representation-theoretic hypotheses hold. Because the reverse implication is already known for totally real fields, the result upgrades rank-one-plus-finite-Sha to an actual equivalence with analytic rank one. This is the first such converse established for a number field beyond the rationals.","feed_headline":"Rank 1 plus finite Sha forces order-one L-vanishing","feed_subtitle":"For elliptic curves over real quadratic fields, algebraic rank one then implies the complex L-function vanishes simply at s=1.","key_machinery":"The carrying object is the Hida family, a p-adic analytic family of Hilbert modular forms that specializes to the eigenform attached to E. Associated to it are a four-variable p-adic L-function and a Selmer group whose dual has a characteristic ideal. The argument is a four-inequality chain: the Iwasawa main conjecture divisibility bounds the order of the p-adic L-function above by a Selmer length; a second-derivative formula bounds that order below by four; Selmer complexes with a p-adic weight pairing, plus Kummer theory, pin the relevant Selmer length to two; and a comparison lemma shows the imprimitive Selmer length is at most the strict length plus two. These force the p-adic order to be exactly four, which is equivalent to order-two vanishing of the complex L-function over a chosen quadratic extension, and the desired order-one vanishing over F follows from a suitable choice of that extension.","core_discovery":"The central discovery is a conditional proof that, under the hypotheses (irred) and (MML), the conditions rank_Z E(F)=1 and #III(E/F)_{p^\\infty}<\\infty imply ord_{s=1}L(E/F,s)=1. The proof moves to a carefully chosen imaginary quadratic extension K/F, proves that the central critical p-adic L-function of the associated Hida family vanishes to order exactly four at weight 2, translates this into order-two vanishing of the complex L-function over K, and then uses the factorization L(E/K,s)=L(E/F,s)L(E^K/F,s) together with a nonvanishing twist to conclude order-one vanishing over F. The paper also applies this to elliptic curves over Q, removing a technical hypothesis from an earlier p-converse theorem.","pith_inferences":["Inference: The same proof scheme should adapt to any totally real field satisfying Leopoldt's conjecture and modularity, so the real-quadratic restriction is probably not essential.","Inference: The p-adic weight pairing constructed here might give a route to rank-one p-converse statements for curves with additive or good reduction at p, once the local pairings are replaced appropriately.","Inference: The condition (MML) is the real bottleneck; if minimal modular liftings are proven to exist in greater generality, the theorem immediately covers all such curves without further change."],"forward_implications":["If the theorem is correct, then for every curve satisfying the hypotheses, rank-one plus finite p-primary Sha is equivalent to ord_{s=1}L(E/F,s)=1, not merely implied by it.","The proof yields a precise computation: the central critical p-adic L-function has order exactly 4 at weight 2 for the auxiliary quadratic extension K.","Over Q, the result removes the auxiliary-prime hypothesis from earlier p-converse theorems for split multiplicative primes p>5: rank-one plus finite Sha suffices to conclude analytic rank one.","The strategy provides a template for extending p-converse theorems to totally real fields when the corresponding modularity and Leopoldt-type hypotheses hold."],"supporting_citations":[{"why":"Supplies the analytic-to-algebraic direction: order-one vanishing of the complex L-function implies rank one and finite Sha, the converse the paper proves.","marker":"[Zha01]"},{"why":"Establishes the Iwasawa main conjecture divisibility for Hilbert modular forms, giving the first inequality in the proof.","marker":"[Wan15]"},{"why":"Provides the formula for the second derivative of the p-adic L-function at weight 2, yielding the order-at-least-four bound.","marker":"[Mok11]"},{"why":"Develops Selmer complexes and the nondegenerate Cassels-Tate pairing used to compute the length of the Selmer group.","marker":"[Nek06]"},{"why":"Proves the factorization of the p-adic L-function as a product of two two-variable p-adic L-functions.","marker":"[LZ20]"},{"why":"Shows elliptic curves over real quadratic fields are modular, attaching the Hilbert modular form f.","marker":"[FLHS15]"},{"why":"Provides the nonvanishing-of-twists criterion used to select the auxiliary imaginary quadratic field K.","marker":"[FH95]"},{"why":"Gives the parity result that finite Sha and rank one force the global root number to be -1.","marker":"[Nek18]"},{"why":"Supplies the control-theorem step used in deducing the characteristic-ideal inequality at the central critical point.","marker":"[SU14]"},{"why":"Provides the interpolation property for p-adic L-functions of Hilbert modular forms, used in the factorization identity.","marker":"[BH24]"}],"fun_headline_variants":["p-Converse for real quadratic fields: rank one implies L-order one","Rank one elliptic curves over real quadratic fields force simple L-vanishing","Conditional p-converse: rank one over real quadratic field gives L-order one","From Sha finite to L-order one: elliptic curves over real quadratic fields"],"cache_read_input_tokens":37376,"weakest_assumption_plain":"Everything hinges on a technical existence hypothesis called (MML): a certain p-adic modular form must lift the residual Galois representation in a minimal way, and without it the Iwasawa-main-conjecture input cannot be invoked.","fun_headline_variants_meta":{"raw":{"variants":["p-Converse for real quadratic fields: rank one implies L-order one","Rank one elliptic curves over real quadratic fields force simple L-vanishing","Conditional p-converse: rank one over real quadratic field gives L-order one","From Sha finite to L-order one: elliptic curves over real quadratic fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000534,"raw_usage":{"total_tokens":2540,"prompt_tokens":891,"completion_tokens":1649,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":1577}},"tokens_in":507,"tokens_out":1649,"duration_ms":14643,"temperature":1.0,"reasoning_tokens":1577,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:53:25.631918+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the paper's explicit example, the base change of the elliptic curve X0(11) to Q(\\sqrt{2}) with p=11, and compute the Hasse-Weil L-function near s=1; the theorem predicts L(E/F,1)=0 with a nonzero first derivative, so a computed order of vanishing different from 1 would refute the theorem. Alternatively, compute the order of vanishing at k=2 of the central critical p-adic L-function L_cc_p(f/K,k) for the auxiliary K, which the proof forces to be exactly 4.","supporting_citations":[],"review_version":1}