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Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2507.18574 v3 pith:KQQKAVWH submitted 2025-07-24 math.NT

Asuka Shiga This is my paper · ORCID
classification math.NT MSC 11G0511G40
keywords ellipticcurvesBSDinvariantsquadratictwistsTate-ShafarevichgroupSelmergroups2-isogenyKodairasymbolsj-invariant
topics Birch and Swinnerton-Dyer
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

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.

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 (2)
  1. [Abstract and Theorem 5.8] 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.
  2. [Theorem 5.8 proof] 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.
minor comments (6)
  1. [Section 3, Proposition 3.2 and Theorem 3.4] 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'.
  2. [Section 3, Proposition 3.5] 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.
  3. [Section 5, Example 5.4 and Lemma 6.1] 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.
  4. [Section 6, Proposition 6.3] 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.
  5. [Section 6.2, Example 6.5] 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'.
  6. [Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the BSD-twin construction rests on independent distribution theorems, local invariant formulas, and explicit Sage/LMFDB computations.

full rationale

The derivation chain is not circular. The equality BSD(E_1^D/Q)=BSD(E_2^D/Q) is never assumed as an input; it is obtained from Proposition 5.3, whose hypotheses are the existence of a balanced 2-isogeny, equality of the 2-primary parts of the Mordell--Weil and Tate-Shafarevich groups, even analytic rank, and matching local data. These inputs are supplied independently: the balanced isogeny and Tamagawa equality for the starting pair are proved in Proposition 5.6; the simultaneous Selmer trivialization comes either from Smith's equidistribution theorem (Theorem 3.4, an external result) or from the explicit 2-descent in Proposition 6.3, and the paper explicitly notes that the explicit descent alone would suffice; Dokchitser--Dokchitser and Kloosterman--Schaefer formulas are external local invariant tools; and the m=1 pair 38025.ck1/ck2 is checked by Sage and LMFDB. The uniqueness classification in Theorem 1.4 is cited from [1], not from the author's own prior work. The only self-citation is [16], a pointer for possible 3-torsion in Sha in Example 6.5, and it is not used in any proof. The real limitation is not circularity: Theorem 5.8 is conditional on rank_an(E_m/Q) being even, verified only for six values of m, so the infinite pairwise-distinct-j family in the abstract is not unconditional; however, this is an unproved hypothesis, not a reduction of the conclusion to its own assumption.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard external theorems and on computational inputs from LMFDB and Sage for the base pair. No free parameters are fitted; the main risk is in the applicability of the quoted equidistribution theorem and in the unproven infinitude of m satisfying the parity hypothesis.

assumptions (8)
  • standard math Faltings's isogeny theorem: two elliptic curves over Q with the same L-function are isogenous.
    Used in Section 1.1 to restrict BSD twins to an isogeny class; a foundational theorem.
  • standard math Smith's equidistribution theorem (Theorem 1.10 of [20]), restated as Theorem 3.4, for the joint density of dim Sel_phi and dim Sel_hatphi over quadratic twists of a balanced 2-isogeny.
    Load-bearing for Proposition 3.5: it provides infinitely many D with Selmer dimensions equal to 1. If the theorem's hypotheses fail for the constructed E_m, the infinite twists are not obtained.
  • standard math Monsky's theorem (Theorem 1.5 of [11]): when the 2-infinity Selmer rank is zero the root number is +1, so analytic rank is even.
    Used in Proposition 3.5 and implicitly in Proposition 6.3 to convert X[2]=0 and rank 0 into the even analytic rank needed for real-period equality.
  • standard math Dokchitser-Dokchitser criterion (Theorem 8.2 of [4]): for p-isogenous curves, equality of real periods is equivalent to a parity congruence involving Tamagawa ratios and analytic rank.
    Core to Theorem 4.3, which is used repeatedly to promote local equalities to real-period equality.
  • standard math Schaefer's local formula (Lemma 3.8 of [14]): c_{E2/Qp} / c_{E1/Qp} = (1/2) #W_p(phi,D) for odd primes, used to transport Tamagawa equalities through twists.
    Essential in Theorem 5.7 and Proposition 6.3 to control Tamagawa numbers at primes of bad reduction.
  • standard math Gouvea-Mazur theorem (Proposition 1 of [5]): only finitely many quadratic twists have a torsion point of order greater than 2.
    Used in Theorem 3.3 and Theorem 5.7 to discard exceptional twists while preserving the infinitude of D.
  • standard math Kloosterman-Schaefer Selmer ratio formula (Theorem 1 and Remark 1 of [8]).
    Used in Theorem 5.8 and Lemma 6.1 to compute tau_1 = 1 from period, Tamagawa, and torsion factors.
  • standard math Barrios et al. [1]: classification of 2-isogenous discriminant twins, with uniqueness of (4225.h1,4225.h2) up to quadratic twist.
    Used to identify the m=1 pair as the unique starting pair for the stronger local-indistinguishability result.

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Pith. "Pith review of Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants." pith.science (2026). https://pith.science/paper/KQQKAVWH

@misc{pith2026250718574,
  author       = {Pith},
  title        = {Pith review of: Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQQKAVWH}},
  note         = {Machine review of arXiv:2507.18574}
}
abstract

Let \(E/\mathbb{Q}\) be an elliptic curve. The Birch and Swinnerton--Dyer (BSD) conjecture relates the leading coefficient of the Taylor expansion of the \(L\)-function of \(E/\mathbb{Q}\) at \(s=1\) to arithmetic invariants of \(E\), such as its Mordell--Weil group, its Tate--Shafarevich group, its Tamagawa numbers, its regulator, and its real period. We call two non-isomorphic elliptic curves over \(\mathbb{Q}\) BSD twins if they have the same \(L\)-function and the same arithmetic data underlying the BSD invariants appearing in the BSD conjecture, with the Mordell--Weil group and the Tate--Shafarevich group compared as groups. We exhibit a family of BSD twins for which the corresponding pairs of \(j\)-invariants are pairwise distinct. We further prove that, even after imposing equality of the Kodaira symbols at every prime and equality of the minimal discriminants, infinitely many BSD twins still exist.

Figures

Figures reproduced from arXiv: 2507.18574 by the authors.

Figure 1
Figure 1. Theorem 5.7 acts as a twisting machine. Applied to the abundant family in Example 1.2, it produces BSD twins over Q. Applied to the very rare discriminant twins in Example 5.4, it produces infinitely many pairs over Q for which the BSD invariants, the Kodaira symbols, and the minimal discriminants all coincide. Finally, we explain the related work relevant to the present study. In the high-dimensional case, Jamie Be… view at source ↗

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Works this paper leans on

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