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On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture

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Pith's one-line read This paper identifies 36,687 elliptic curves whose infinite quadratic-twist families all satisfy the Birch–Swinnerton-Dyer conjecture unconditionally, and documents a systematic positive bias in the resulting Tate–Shafarevich orders.

desk verdict Useful algorithmic extension of BSTW with a real dataset, but the 'all' claim is overstated and Theorem 2.2 needs a proof. read the letter →

arxiv 2601.16044 v3 pith:AJJJ3VUJ submitted 2026-01-22 math.NT

Barinder S. Banwait , Xiaoyu Huang This is my paper
classification math.NT MSC 11G0511Y1614Q05
keywords ellipticcurvesquadratictwistsBirch-Swinnerton-DyerconjectureTate-ShafarevichgrouprankzeroalgorithmicnumbertheoryGaussiandistributionconductorbound
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 tries to establish that a specific, machine-checkable list of 36,687 elliptic curves of conductor at most 500,000 has the property that each curve admits an explicit infinite family of quadratic twists for which the full Birch–Swinnerton-Dyer conjecture is unconditionally true. It does so by translating the hypotheses of recent theorems into an explicit two-stage algorithm and running it over the entire database of such curves, extending earlier small examples and correcting a prior list. If correct, the result provides a large dataset in which the order of the Tate–Shafarevich group is known unconditionally for entire infinite twist families. The paper then uses that dataset to test the Gaussian prediction for Tate–Shafarevich orders: generic twists converge to the predicted bell curve, while the BSD-satisfying subfamily shows a systematic positive bias and a bimodal shape.

What carries the argument

The central mechanism is a two-stage filtering algorithm. Algorithm 1 takes an elliptic curve and checks sufficient conditions: squarefree non-prime conductor, optimality, odd Manin constant, rank zero, absence of rational p-isogenies for relevant primes, the 2-adic valuation of the algebraic L-value, and—in the Z/2Z torsion case—conditions on the 2-isogenous curve and its 2-part of the Tate–Shafarevich group. Algorithm 2 takes an accepted curve and a squarefree twist d and verifies local congruence and splitting conditions (e.g. d ≡ 1 mod 4 or 8, inertness of twist primes in the 2-division field, and Legendre-symbol conditions at the conductor) that force the twist to have rank zero and to

What would settle it

Take any curve accepted by the paper and any twist d it lists; compute the algebraic and analytic ranks and the size of the 2-primary Selmer group of E^d. If any such twist has positive rank, or if its Selmer/Shafarevich group size disagrees with the BSD formula, the claimed certificate is wrong. Conversely, exhibit a semistable non-CM curve of conductor below 500,000 that satisfies all the p-part hypotheses of the assembled theorems but is not in the output, which would refute the 'all' phrasing.

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Extended reading notes

Core claim

Theorem 1.1 asserts that for every curve in the computed set C, there exists an explicit infinite family of quadratic twists {E^d} such that each twist satisfies the full Birch–Swinnerton-Dyer conjecture. The set C contains 36,687 curves, about 0.1% of all elliptic curves of conductor below 500,000. For each accepted curve the algorithm certifies analytic rank zero, verifies the odd-prime and 2-primary parts of BSD from an aggregation of published theorems, and then outputs the admissible twist discriminants in a range. The paper further computes the unconditionally known Tate–Shafarevich orders for these twists and compares their normalized distribution with the Gaussian law predicted by ra

Load-bearing premise

The argument stands or falls on Theorem 2.2 being a complete and correct statement of the 2-part BSD conditions; its rank-zero requirement rests on a private communication, and the full rational 2-torsion case is acknowledged missing.

Editorial extensions

If this is right

  • For each of the 36,687 listed curves, every twist produced by Algorithm 2 has proven analytic rank zero, provable p-part of BSD for all primes, and an unconditionally known order of the Tate–Shafarevich group.
  • The earlier known examples up to conductor 150 are extended to 36,687 curves up to conductor 500,000, and four previously listed curves are removed because the current theorems do not cover their rational-2-torsion case.
  • The numerical distribution of normalized Tate–Shafarevich orders for generic twists of the example curve approaches the standard Gaussian as the discriminant bound grows, as measured by two standard goodness-of-fit distances.
  • The 1,008 BSD-satisfying twists of the example curve within the tested range show a significant positive bias and bimodality relative to that Gaussian prediction, indicating that the sieve conditions alter the arithmetic distribution.

Reading between the lines

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

  • If the output list is complete, the same sieve could be rerun at larger conductor bounds; the roughly 0.1% acceptance rate suggests the count will grow but remain sparse.
  • The observed bias is plausibly explained by the imposed local conditions (all twist primes 1 mod 4, real quadratic fields, character fixed at the conductor); an extension would be to fold these restrictions into the moment computation and test whether a shifted Gaussian emerges.
  • The 36,687 figure is conditional on the current set of 2-part theorems; proving analogous results for the full rational 2-torsion case would likely enlarge the accepted set, so 'all' should be read as 'all within the present theorem kit'.
  • The algorithm's structure transfers to other arithmetic families, e.g. curves with prescribed torsion or in fixed isogeny classes, so the same approach could certify infinite BSD-twist families elsewhere.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper builds on Burungale–Skinner–Tian–Wan's theorem producing infinitely many quadratic twists satisfying the full BSD conjecture. It encodes the hypotheses of that theorem and of earlier works on the 2-part of BSD into two algorithms, runs them on LMFDB curves of conductor below 500,000, and reports a set of 36,687 elliptic curves for which it claims (Theorem 1.1) there are explicit infinite families of BSD-satisfying twists. It also tests the Radziwiłł–Soundararajan prediction numerically on generic twists of 46a1 and on the 1008 BSD-verified twists up to |d| ≤ 300,000, reporting a systematic positive bias in the latter subfamily.

Significance. If fully correct, the paper would provide a large, reproducible database of non-CM elliptic curves whose twists provably satisfy the full BSD conjecture, complementing the small list in [4, Example 1] and enabling numerical evidence that is independent of assuming BSD. The algorithms are concrete, the code is public, and the data file is linked, which are strengths. However, the central theorem depends on Theorem 2.2, a self-assembled 2-part criterion whose proof rests on unstated, privately communicated assumptions; and the abstract's 'all' wording overstates what is derived, since the algorithm only extracts curves satisfying sufficient conditions. The statistical section also has unresolved selection-bias issues.

major comments (4)
  1. [Section 2.2, Theorem 2.2 and its proof] This is the load-bearing result: Algorithm 1, Step 8, invokes Theorem 2.2 to accept every one of the 36,687 curves. The proof, however, adds a rank-0 hypothesis that the cited theorems [5, 22] do not state, justifying it only by modular-symbol heuristics and 'confirmed by Zhai in private communication'. This is not independently verifiable. Similarly, condition (3c) — E'(Q)[2] ≅ Z/2Z and X(E')[2] = 0 — is said to come from Remark 1.3 of [5], not from a theorem statement. The authors should either supply a complete proof of Theorem 2.2 with all hypotheses, or clearly state it as a conditional aggregation and identify which of the listed curves depend on the unverified hypotheses.
  2. [Abstract and Remark 2.3] The abstract claims the algorithm identifies 'all elliptic curves E of conductor at most 500,000 that admit infinitely many quadratic twists satisfying the strong BSD conjecture'. This is not established by the paper: Theorem 2.2 does not cover the full rational 2-torsion case (as Remark 2.3 admits), and the algorithm checks sufficient conditions, not necessary ones. At most the paper identifies all curves satisfying the explicit hypotheses of Algorithms 1–2. The wording must be corrected, and the precise sense of 'identifying all' must be stated in the introduction and abstract.
  3. [Section 3.2, Figure 3.1 and the exclusion criterion] The text says Figure 3.1 excludes 472 values 'where the computation for the order of X(E)an took longer than 10 seconds and so was abandoned'. If those slow computations are not a random sample — e.g., if they correspond to large analytic orders, small regulators, or particular residue classes — then the K–S and Wasserstein distances in Figure 3.2 are computed on a biased subsample. The authors should either report the number and distribution of excluded values, give bounds on their effect, or rerun with a complete computation for a smaller X.
  4. [Section 3.3, Figure 3.3 and the claimed positive bias] The deviation from the RS prediction is based on only N=1008 twists of the single curve 46a1. The text calls the distribution 'bimodal' and 'shifted to the right', but no statistical test, confidence interval, or comparison with a modified RS model with restricted parameters is provided. Since the restrictions (p ≡ 1 mod 4, d ≡ 1 mod 8, (d/23)=1) are exactly the kind that alter µ and σ, the authors should compute the conditional mean/variance predicted by the RS method for this sieve, or at least provide a formal goodness-of-fit assessment. Without this, the 'systematic positive bias' claim is only an observation for one curve and one truncation.
minor comments (6)
  1. [Section 2.2, proof of Theorem 2.2] Typo: 'we therefore state provide the following result' should be 'we therefore state the following result' or 'provide'.
  2. [Remark 2.3] The text refers to 'Lemma 2.3' and 'Lemma 2.4' in Section 2.4, but these are Remarks 2.3 and 2.4. Please correct the cross-references.
  3. [Algorithm 2, line 6] The condition 'if p|N is odd, then (Δ_{Q(√d)}/p) = 1' uses notation that is not defined and appears to depend on p, but the Legendre-symbol expression is not written explicitly. Clarify whether p runs over odd prime divisors of N and define the discriminant symbol.
  4. [Section 2.4, bullet list comparison] The comparison with [4, Example 1] is clear, but the reason the four curves 62a1, 66b1, 105a1, 141c1 fail is stated only in terms of ord_2(L_alg(E,1)) = −2. It would help to state the exact theorem in [5] requiring ord_2 = −1.
  5. [Section 3.1] The definition of µ(E) and σ(E) uses the notation c(g) = 1 + |Fix(g)|, but Fix(g) is defined as 'the number of fixed points of g acting on the roots of f'. This should be clarified: does it count only fixed roots, or the identity fixed points? Also, the formula for σ(E)^2 appears to lack the subtraction of (log c(g))^2 mean term; please check against [16].
  6. [General] The paper states that the output file [1] is a GitHub repository. Since Theorem 1.1 cites this repository, the authors should specify the exact commit/version and provide a checksum or hash of the output file, making the computational claim reproducible and verifiable.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the BSD-twist list is produced by applying external theorems to the LMFDB database, and the statistical parameters are computed, not fitted.

full rationale

The paper's derivation chain is not circular. Theorem 1.1 depends on the authors' own output file [1], but that file is a computational artifact generated by Algorithms 1 and 2, which in turn implement external results (Burungale-Skinner-Tian-Wan, Skinner-Urban, Zhai, Cai-Li-Zhai). The existence of infinitely many BSD-satisfying twists for each listed curve follows from those external theorems plus explicitly executable local checks; it is not assumed as an input. The self-citation to [1] is therefore code-reproducible evidence rather than load-bearing circular reasoning. The statistical section likewise contains no fitted-input-called-prediction step: the Radziwill-Soundararajan parameters mu(E) and sigma(E) are computed from the Galois action of the 2-division field, not tuned to match the observed distribution, and the comparison between generic twists and the BSD subfamily is empirical. The paper does contain acknowledged caveats: Theorem 2.2's proof adds a rank-0 hypothesis supported only by a private communication, Remark 2.3 concedes the full rational 2-torsion case is missing, and Section 2.4(2) notes four curves are not included. These are correctness and completeness risks about the aggregation of external hypotheses, not instances where a prediction reduces by construction to its own input. No equation is defined in terms of the claimed output, and no parameter is fitted then renamed as a prediction. Thus the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted; µ(E) and σ(E) are computed from the Galois action. The central claim inherits its content from external Iwasawa-theory theorems and the unproved 2-part aggregation, which is why the axioms listed above are the real price of admission.

assumptions (3)
  • domain assumption Burungale–Skinner–Tian–Wan [4, Thm 10.12] together with [19,18,5,22] correctly prove the p-part of BSD for rank-zero non-CM twists satisfying the listed local conditions.
    The algorithm is a direct encoding of these external theorems; the paper does not re-prove them.
  • ad hoc to paper The 2-part conditions stated in Theorem 2.2, including the added rank-0 requirement and the Z/2Z case hypotheses, are jointly sufficient for the 2-part BSD of E^d; the rank-0 requirement is justified by private communication rather than a published statement.
    This is paper-specific aggregation; a failure here would invalidate Algorithm 1's acceptance tests.
  • domain assumption The LMFDB data (conductor, minimal discriminant, Manin constant, algebraic L-values, torsion) are accurate for all curves with conductor ≤500,000.
    The list is only as reliable as the database it scans; the paper gives no independent audit of LMFDB entries.

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Pith. "Pith review of On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture." pith.science (2026). https://pith.science/paper/AJJJ3VUJ

@misc{pith2026260116044,
  author       = {Pith},
  title        = {Pith review of: On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AJJJ3VUJ}},
  note         = {Machine review of arXiv:2601.16044}
}
abstract

Recent work of Burungale-Skinner-Tian-Wan established the first infinite families of quadratic twists of non-CM elliptic curves over $\mathbb{Q}$ for which the strong Birch-Swinnerton-Dyer (BSD) conjecture holds. Building on their results, we encode the required hypotheses into an explicit algorithm and apply it to the database of elliptic curves in the $L$-functions and Modular Forms Database (LMFDB), identifying all elliptic curves $E$ of conductor at most $500{,}000$ that admit infinitely many quadratic twists satisfying the strong BSD conjecture. Our computations provide certain numerical evidence for a conjecture of Radziwi{\l}{\l} and Soundararajan predicting Gaussian behavior in the analytic order of the Shafarevich-Tate group, while also observing a systematic positive bias within the BSD-satisfying subfamily.

Figures

Figures reproduced from arXiv: 2601.16044 by the authors.

Figure 3
Figure 3. displays the results for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 3.1
Figure 3.1. displays the results for X = 100,000 (excluding 472 values where the computation for the order of X(E) an took longer than 10 seconds and so was abandoned). Kernel density estimates for X ≤ 100,000 show certain agreement with N (0, 1), with mild finite-sample skewness [PITH_FULL_IMAGE:figures/full_fig_p006_3_1.png] view at source ↗
Figure 3
Figure 3. illustrates this behavior for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 3.2
Figure 3.2. Figure 3.2: illustrates this behavior for 46a1 as a representative case, where both distance metrics decrease as the discriminant bound grows to X = 100,000. (a) KS distance (b) Wasserstein distance [PITH_FULL_IMAGE:figures/full_fig_p007_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: Distribution of normalized Zd for 1008 of the the unconditionally BSD-satisfying twists of 46a1. The observed distribution (blue) is clearly shifted to the right of the generic Radziwi l l–Soundararajan prediction (black dashed). This shift reflects the bias introduc…

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