{"id":"241355da-b47e-457c-b81c-13cd49ad7a2b","arxiv_id":"2507.18574","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"There exist infinitely many pairs of non-isomorphic elliptic curves over Q with identical BSD data, and this remains true when equality of Kodaira symbols and minimal discriminants is imposed.","lead":"This paper constructs infinitely many pairs of non-isomorphic elliptic curves over the rationals that have identical BSD invariants, including the same L-function, Mordell-Weil group, regulator, real period, Tamagawa numbers, and Tate-Shafarevich group. It also shows that such pairs survive even when the Kodaira symbols at all primes and the minimal discriminants are forced to agree.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.9's proof is coherent; the real gap is the unproved even-analytic-rank hypothesis needed for the abstract's infinite distinct-j family.","rationale":"I agree with the reader's conditional verdict, but the most load-bearing concern is not the applicability of Smith's equidistribution theorem to the m = 1 pair. The explicit Proposition 6.3 provides a descent-based proof of the simultaneous trivialization of the 2-part of the Tate-Shafarevich groups for an infinite family of twists, independent of Smith's theorem, so Theorem 5.9 has a second, self-contained route. The genuine soft spot is the parity hypothesis for infinitely many starting pairs in Theorem 5.8, which is needed for the abstract's first headline claim: infinitely many BSD twins with pairwise distinct j-invariant pairs. The paper checks only six values of m and gives no argument that rank_an(E_m) is even for infinitely many m in the Legendre-symbol family. Since the theorem is stated conditionally, it is not formally false, but the abstract overstates the result by presenting the infinite distinct-j family as established. This matches the reader's secondary concern and supports a conditional rather than unconditional acceptance. The m = 1 construction leading to Theorem 5.9 appears sound: local invariants are checked, the explicit D family is infinite, and the Kodaira/discriminant equality follows from the stated local theorems. My recommendation is therefore to keep the reader's verdict unchanged rather than to move to accept or reject.","tokens_in":18744,"tokens_out":54367,"duration_ms":553241,"concrete_test":"Compute the root number ω(E_m) for the family E_m of Example 1.2 for primes m with (390/m) = -1, using local root numbers at 2, 3, 5, 13, m and ∞. Express ω(E_m) as a Dirichlet character in m and identify an arithmetic progression of m on which ω(E_m) = +1, which would imply rank_an(E_m) is even, conditional on the known parity theorem. A Sage or Pari script checking all primes m ≤ 2000 in the family should confirm the predicted pattern. If no such progression can be found, Theorem 1.3 should be restated as conditional and the abstract's infinite distinct-j claim withdrawn; if a progression is found, the proof can be completed by a short supplementary argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim as advertised in the abstract is that there is an infinite family of BSD twins whose pairs of j-invariants are pairwise distinct. This rests on Theorem 5.8, which is conditional on 'rank_an(E_m/Q) ≡ 0 mod 2'. The paper verifies this parity condition only for m = 7, 19, 23, 37, 43, 53 and supplies no root-number computation or L-function calculation showing that it holds for infinitely many m satisfying (390/m) = -1. Without infinitely many such m, the set of starting pairs is finite, so the claimed infinite family with pairwise distinct j-invariant pairs is not established unconditionally. The other advertised theorem, Theorem 5.9, is not affected: it uses only the m = 1 pair, and the chain through Proposition 6.3 (or through Theorem 5.7 plus Smith's theorem) appears internally consistent. The Smith-theorem step is a black box, but the paper states that Proposition 6.3 alone suffices for Theorem 5.9, and the explicit 2-descent in that proposition independently supports the m = 1 construction. I therefore do not regard the Smith citation as the primary soft spot; the missing parity proof for infinitely many m is the concrete load-bearing gap in the paper's headline claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the notion of BSD twins: pairs of non-isomorphic elliptic curves over Q with identical BSD data (L-function, Mordell–Weil group, regulator, real period, Tamagawa numbers, and Tate–Shafarevich group, with groups compared as abstract groups). The main results are: (i) a theorem (Thm 5.8 = Thm 1.3) asserting that, for each m in a family of primes with (390/m)=-1 and with even analytic rank of E_m, there are infinitely many quadratic twists giving BSD twins, and that the j-invariant pairs are pairwise distinct as m varies; and (ii) an unconditional theorem (Thm 5.9 = Thm 1.5) asserting the existence of infinitely many BSD twins that additionally share the same Kodaira symbols at every prime and the same minimal discriminant. The proof strategy combines a 'twisting machine' that simultaneously trivializes the 2-primary Selmer groups, using Smith's distribution theorem for the general family and an explicit 2-descent for the m=1 case.","tokens_in":18875,"tokens_out":32958,"duration_ms":304887,"significance":"If the results hold, Theorem 5.9 shows that the full set of BSD data, even together with all local reduction data, does not characterize an elliptic curve over Q. This is a striking and valuable addition to the literature on arithmetic equivalence. The paper is largely self-contained for the unconditional result: the m=1 pair is explicitly computed, the local conditions are checked in detail, and the reliance on LMFDB/Sage is limited to a few finite verifications that are clearly identified. The proof of the conditional Theorem 5.8 is coherent given Smith's theorem and the parity hypothesis. The main weakness is that the headline claim of an infinite family with pairwise distinct j-invariants is not established, because the parity hypothesis is verified only for finitely many m.","major_comments":[{"comment":"The abstract states: 'We exhibit a family of BSD twins for which the corresponding pairs of j-invariants are pairwise distinct.' This claim is not unconditionally established. Theorem 5.8 is conditional on the hypothesis rank_an(E_m/Q) ≡ 0 mod 2, which is verified only for the finite set m = 7, 19, 23, 37, 43, 53. Since quadratic twisting preserves j-invariants, the pairwise distinctness over infinitely many pairs requires infinitely many m satisfying this parity condition. The manuscript supplies no proof or evidence that the parity holds for infinitely many m. The abstract should be revised to state the conditional nature of this family, and the unconditional Theorem 5.9 (which gives infinitely many pairs but with a fixed j-pair) should be clearly separated as the unconditional headline result.","section":"Abstract and Theorem 5.8"},{"comment":"For the six listed values of m, the paper does not document how the condition rank_an(E_m/Q) ≡ 0 mod 2 is verified. No root-number computation, L-value calculation, or reference is given. Since this condition is genuinely load-bearing for the infinite distinct-j family, the authors should either provide the relevant computations or explicitly state that these are isolated numerical checks. More importantly, the question of whether rank_an(E_m/Q) is even for infinitely many primes m with (390/m) = -1 is not addressed anywhere in the manuscript; without such an infinitude statement, the asserted 'family' of BSD twins with pairwise distinct j-invariants is only a finite collection of examples.","section":"Theorem 5.8 proof"}],"minor_comments":[{"comment":"The notation 'S := {v : v|2ΔE1∞}' is nonstandard and should be written more clearly, for example as 'the set consisting of the primes dividing 2ΔE1 together with the infinite place'.","section":"Section 3, Proposition 3.2 and Theorem 3.4"},{"comment":"The condition 'D ∈ (Q_v^×)^2 for every v|2ΔE1ΔE2∞' is used to conclude gcd(D,2ΔE1ΔE2)=1, D>0, and D≡1 mod 8. It would help the reader if this inference were stated explicitly before it is used.","section":"Section 3, Proposition 3.5"},{"comment":"The key input that Sel_2(E_i) ≅ Z/2Z, rank(E_i/Q) = 0, and rank_an(E_1/Q) = 0 for the base pair (38025.ck1, 38025.ck2) is asserted via 'a Sage computation' without code or output. Since this base case is essential for Theorem 5.9, please include the computation or a precise, checkable reference.","section":"Section 5, Example 5.4 and Lemma 6.1"},{"comment":"After establishing X(E_i^D/Q)[2]=0 and rank(E_i^D/Q)=0, the proof does not explicitly repeat the argument from Proposition 3.5 that rank_an(E_i^D/Q) is even via the root number. A short sentence citing the same theorem (Monsky, [11]) would make the deduction transparent.","section":"Section 6, Proposition 6.3"},{"comment":"The notation 'X(E17_1/Q)' is confusing; it should be written as 'X(E_1^{17}/Q)' or 'X(E_1^D/Q) for D = 17'.","section":"Section 6.2, Example 6.5"},{"comment":"There are minor typographical issues: 'rank an' should be 'rank_an', the symbol 'Ω E' has an extra space, and the set notation in Proposition 3.2 should be cleaned up. A careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a solid unconditional theorem (Theorem 5.9) and an interesting but conditional result (Theorem 5.8). The abstract systematically overstates the strength of the results by not mentioning the parity hypothesis for the infinite distinct-j family. I recommend a major revision in which the conditional nature of Theorem 5.8 is made explicit, the parity verification for the finite list is documented, and the unconditional theorem is foregrounded. The mathematical core appears sound, so rejection is not warranted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result. The paper proves there are infinitely many non-isomorphic elliptic curves over Q with identical BSD data, identical Kodaira symbols everywhere, and identical minimal discriminants. That is a clean negative answer to the question of whether BSD data characterize a curve, and I don't see a load-bearing flaw in the proof of Theorem 5.9.\n\nWhat's new: the 'BSD twin' framing, the 2-isogeny twisting machine that simultaneously trivializes the 2-parts of both Tate-Shafarevich groups, and the observation that the unique 2-isogenous discriminant twin pair (4225.h1/h2) can be twisted to fix the parity issue (the original pair has odd analytic rank, so the twisted pair 38025.ck1/ck2 is used). The explicit 2-descent in Section 6 gives a concrete infinite family of D's for the m=1 pair, which makes Theorem 5.9 independent of Smith's equidistribution theorem. That matters, because Smith's theorem is quoted as a black box; the explicit descent is checkable.\n\nThe soft spot is real but localized. The abstract and Theorem 5.8 promise an infinite family of BSD twins whose pairs of j-invariants are pairwise distinct as the starting pair varies. The proof needs rank_an(E_m/Q) even for infinitely many m, and the paper only verifies m = 7, 19, 23, 37, 43, 53. No root-number argument or L-function computation is supplied to show the parity condition holds infinitely often. So that part of the headline is not established. The paper should either prove the parity condition for infinitely many m (plausible, since the root number is computable) or rewrite the abstract to state the unconditional Theorem 5.9 and demote the pairwise-distinct family to a conditional result. Minor: computations are described but code not shipped; the LMFDB and Sage claims are specific enough to reproduce, though.\n\nI agree with the stress-test note: the Smith step is not the primary weak point. The concrete gap is the missing infinitude of even analytic rank for the E_m family. The reader's conditional verdict is about right.\n\nWho this is for: anyone working on elliptic curves, BSD invariants, or isogeny classes. It deserves a serious referee. My recommendation: send it to peer review, with a request that the parity question be addressed head-on.","headline":"Main unconditional theorem is sound and answers the rigidity question negatively, but the abstract's stronger pairwise-distinct family rests on an unproved parity condition for infinitely many m.","tokens_in":19480,"tokens_out":2320,"would_cite":true,"duration_ms":24262,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11G40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the BSD invariants—the L-function, Mordell–Weil group, regulator, real period, Tamagawa numbers, and Tate–Shafarevich group—do not determine an elliptic curve over Q: infinitely many non-isomorphic pairs share them…","keywords":["elliptic curves","BSD invariants","quadratic twists","Tate-Shafarevich group","Selmer groups","2-isogeny","Kodaira symbols","j-invariant"],"falsifier":"Compute, for a handful of square-free D satisfying D≡1 mod 8, gcd(D,2·3·5·13)=1, and (65/p)=−1 for every p|D (e.g., D=17, 41, 73), the 2-Selmer groups of the twisted pair E_1^D and E_2^D; Proposition 6.3 predicts both are of dimension 1 and the 2-parts of the Tate–Shafarevich groups vanish, so the first D for which either Selmer group has a different dimension would refute the explicit construction and undermine the density input.","tokens_in":18429,"feed_emoji":"","tokens_out":9858,"duration_ms":88894,"temperature":0.7,"pith_summary":"The paper asks whether the data appearing in the Birch–Swinnerton-Dyer conjecture uniquely characterize an elliptic curve over Q, and answers no. It constructs infinitely many pairs of non-isomorphic curves with the same L-function, the same Mordell–Weil group, the same regulator, the same real period, the same Tamagawa numbers, and isomorphic Tate–Shafarevich groups, yet with different j-invariants. The strongest theorem goes further: even after fixing the Kodaira symbol at every prime and the minimal discriminant, infinitely many such pairs remain. The construction is a twisting machine: take a fixed 2-isogenous pair, vary the quadratic twist, and use a density theorem for Selmer groups to force the 2-primary parts of the Tate–Shafarevich groups to vanish simultaneously, after which a general criterion promotes this to full equality of BSD data.","feed_headline":"Elliptic curves share all BSD data but not j-invariants","feed_subtitle":"Even identical reduction types and minimal discriminants fail to tell the curves apart.","key_machinery":"A 2-isogeny φ:E1→E2 over Q is balanced when Q(E1[2])=Q(E2[2]); the paper works with balanced 2-isogenies because for them the Tamagawa ratio τ_D = #Sel_φ(E_1^D)/#Sel_φ̂(E_2^D) is independent of the twist D (Proposition 3.2), and a density theorem from the literature (Theorem 3.4) gives the rate at which both Selmer groups have dimension 1 among twists satisfying certain local conditions. This yields, via Proposition 3.5, infinitely many twists with vanishing 2-primary Tate–Shafarevich groups and rank 0. Proposition 5.3 is then the criterion that turns these conditions, plus equality of Tamagawa numbers or of minimal discriminants, into equality of the full six BSD invariants; the m=1 pair satisfies the discriminant version, so the Kodaira symbols and minimal discriminants match as well.","core_discovery":"The central discovery is stated in Theorem 5.9: there exist infinitely many pairs (E1,E2) of non-isomorphic elliptic curves over Q such that j(E1)≠j(E2), BSD(E1/Q)=BSD(E2/Q), and the Kodaira symbols and minimal discriminants are the same at every prime. The proof exhibits the quadratic twists of a specific pair (the m=1 case of the family E_m: y²=x³+390n x²+1952m²n x, E′_m: y²=x³−195n x²+16·195²n x, with n=m²+64). For every square-free D with gcd(D,2·3·5·13)=1, D≡1 mod 8, and (65/p)=−1 for all p|D, the twisted pair has Sel_φ ≅ Sel_φ̂ ≅ Z/2Z, trivial 2-part of the Tate–Shafarevich groups, rank 0, and even analytic rank; the local conditions then force the BSD data, Kodaira symbols, and minimal discriminants to agree.","pith_inferences":["One might expect that p-isogenies for odd primes p admit an analogue of the balanced 2-isogeny machinery; if so, the same 'twisting machine' could produce BSD twins with Tate–Shafarevich groups whose odd primary parts are nonzero, going beyond the trivial-2-part examples here.","The paper verifies the even-analytic-rank hypothesis only for finitely many m (7, 19, 23, 37, 43, 53); if this parity condition holds for infinitely many m, the abundant family of Theorem 1.3 would yield BSD twins with pairwise distinct j-invariants unconditionally.","The explicit local conditions for D (D≡1 mod 8, gcd(D,2·3·5·13)=1, (65/p)=−1) suggest a purely local characterization of which twists become BSD twins; computationally testing the density of such D in intervals could be a quick sanity check of the equidistribution theorem's predictions."],"forward_implications":["The six-term BSD package is not a complete invariant: infinitely many non-isomorphic curves share the same L-function, Mordell–Weil group, regulator, real period, Tamagawa numbers, and Tate–Shafarevich group.","Adding the Kodaira symbol and minimal discriminant at every prime does not repair the failure; Theorem 5.9 gives infinitely many counterexamples even with these data fixed.","The construction is explicit: for the starting pair in Example 5.4, any square-free D satisfying the stated congruence and Legendre-symbol conditions yields a new BSD twin pair, and the proof does not rely on the BSD conjecture or on finiteness of Tate–Shafarevich groups.","For any prime p, if two p-isogenous curves have isomorphic p-primary Tate–Shafarevich groups, their full Tate–Shafarevich groups are isomorphic (Proposition 5.2), so the equality of the 2-primary parts obtained by the twisting machine is enough to conclude global equality without assuming finiteness."],"supporting_citations":[{"why":"Supplies the equidistribution theorem for Selmer groups of balanced 2-isogenies in quadratic twist families, the density input that produces infinitely many twists with 1-dimensional Selmer groups.","marker":"[20]"},{"why":"Classifies 2-isogenous discriminant twins and identifies the unique pair (up to twist) whose twists are used for Theorem 5.9.","marker":"[1]"},{"why":"Provides the local comparison theorems (real periods, Tamagawa numbers, Kodaira symbols under isogeny) that power Proposition 5.3.","marker":"[4]"},{"why":"Gives the Tamagawa-ratio formula used to show the Selmer dimensions of the two curves are equal in the family.","marker":"[8]"},{"why":"Supplies the product formula for the Tamagawa ratio over local factors, used in Proposition 3.2.","marker":"[7]"},{"why":"Shows that only finitely many twists have torsion points of order greater than 2, used to exclude finitely many bad twists.","marker":"[5]"},{"why":"Relates the 2∞-Selmer rank to the root number, giving the even analytic rank condition in Proposition 3.5.","marker":"[11]"}],"fun_headline_variants":["Infinitely many non-isomorphic elliptic curves share all BSD data","Elliptic doppelgangers: same BSD, different j, infinitely many","Same BSD data, distinct j: infinite elliptic pairs","Endless elliptic twins with identical BSD but different j","Infinitely many elliptic pairs share BSD invariants, not j"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction relies on a cited density theorem for balanced 2-isogeny twists: if that theorem does not apply to the particular 2-isogenous pairs used here, there is no proof that infinitely many suitable twists exist.","fun_headline_variants_meta":{"raw":{"variants":["Infinitely many non-isomorphic elliptic curves share all BSD data","Elliptic doppelgangers: same BSD, different j, infinitely many","Same BSD data, distinct j: infinite elliptic pairs","Endless elliptic twins with identical BSD but different j","Infinitely many elliptic pairs share BSD invariants, not j"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000826,"raw_usage":{"total_tokens":3614,"prompt_tokens":954,"completion_tokens":2660,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":2575}},"tokens_in":570,"tokens_out":2660,"duration_ms":19877,"temperature":1.0,"reasoning_tokens":2575,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:14:59.970809+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a handful of square-free D satisfying D≡1 mod 8, gcd(D,2·3·5·13)=1, and (65/p)=−1 for every p|D (e.g., D=17, 41, 73), the 2-Selmer groups of the twisted pair E_1^D and E_2^D; Proposition 6.3 predicts both are of dimension 1 and the 2-parts of the Tate–Shafarevich groups vanish, so the first D for which either Selmer group has a different dimension would refute the explicit construction and undermine the density input.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies 2-isogenous discriminant twins and identifies the unique pair (up to twist) whose twists are used for Theorem 5.9."},{"cited_title":"Dokchitser and V","cited_arxiv_id":null,"evidence_quote":"Provides the local comparison theorems (real periods, Tamagawa numbers, Kodaira symbols under isogeny) that power Proposition 5.3."},{"cited_title":"Kloosterman and E","cited_arxiv_id":null,"evidence_quote":"Gives the Tamagawa-ratio formula used to show the Selmer dimensions of the two curves are equal in the family."},{"cited_title":"Klagsbrun and R","cited_arxiv_id":null,"evidence_quote":"Supplies the product formula for the Tamagawa ratio over local factors, used in Proposition 3.2."},{"cited_title":"Gouvˆ ea and B","cited_arxiv_id":null,"evidence_quote":"Shows that only finitely many twists have torsion points of order greater than 2, used to exclude finitely many bad twists."},{"cited_title":"Monsky, Generalizing the Birch–Stephens theorem","cited_arxiv_id":null,"evidence_quote":"Relates the 2∞-Selmer rank to the root number, giving the even analytic rank condition in Proposition 3.5."}],"review_version":1}