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arxiv: 2602.19861 · v2 · submitted 2026-02-23 · 🧮 math.NT

Nontrivial torsion in the Tate--Shafarevich group of elliptic curves via visibility and twists

Asuka Shiga This is my paper

Pith reviewed 2026-05-15 20:11 UTC · model grok-4.3

classification 🧮 math.NT MSC 11G0511G40
keywords elliptic curvesTate-Shafarevich groupvisibilityquadratic twiststorsionBirch-Swinnerton-DyerKodaira symbolsadditive reduction
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The pith

Quadratic twists of elliptic curves with additive reduction at odd primes ℓ produce nontrivial ℓ-torsion in their Tate-Shafarevich groups via visibility.

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

The paper establishes that the visibility theorem applies to elliptic curves over the rationals with additive reduction at an odd prime ℓ, yielding quadratic twists E^D whose Tate-Shafarevich groups contain nontrivial elements of order ℓ. This matters because the Tate-Shafarevich group measures the obstruction to the Hasse principle for rational points on elliptic curves, and its torsion structure is typically hard to control explicitly. For the case ℓ=3 the construction produces explicit pairs of non-isomorphic elliptic curves that share the same Birch-Swinnerton-Dyer invariants, Kodaira symbols, and minimal discriminants, yet possess isomorphic Tate-Shafarevich groups each having nontrivial 3-primary part. A sympathetic reader would care because these examples demonstrate that the standard arithmetic invariants do not determine the isomorphism type of Sha.

Core claim

By applying the visibility theorem to elliptic curves E over Q that have additive reduction at an odd prime ℓ, one obtains quadratic twists E^D such that Sha(E^D/Q) contains a nontrivial element of order ℓ. For ℓ=3 this produces concrete pairs of non-isomorphic elliptic curves over Q that agree on Birch-Swinnerton-Dyer invariants, Kodaira symbols, and minimal discriminants, but whose Tate-Shafarevich groups are isomorphic and each contain nontrivial 3-torsion.

What carries the argument

The visibility theorem, which maps certain cohomology classes into the Selmer group of an elliptic curve, applied after a quadratic twist to curves with additive reduction at ℓ.

If this is right

  • Nontrivial ℓ-torsion exists in Sha(E^D/Q) for suitable quadratic twists E^D of elliptic curves with additive reduction at ℓ.
  • For ℓ=3 there exist non-isomorphic elliptic curves over Q that share BSD invariants, Kodaira symbols, and minimal discriminants yet have isomorphic Sha groups with nontrivial 3-torsion.
  • The visibility map after twisting supplies an explicit way to produce torsion in Sha while preserving local reduction types.
  • The construction works for any odd prime ℓ, giving a uniform source of Sha torsion across many primes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same twisting-plus-visibility method could be tested on curves with multiplicative reduction to see whether the additive-reduction hypothesis is essential.
  • The pairs of curves with matching invariants but isomorphic Sha could serve as test cases for conjectures relating Sha to the L-function.
  • The technique might generalize to produce examples in which Sha torsion affects the parity of the analytic rank in controlled ways.

Load-bearing premise

The visibility theorem applies to the chosen elliptic curves with additive reduction at ℓ and the quadratic twist produces a curve for which the visibility map detects the torsion element.

What would settle it

For one of the explicit ℓ=3 examples, compute the 3-Selmer rank and the order of Sha and verify that the 3-primary part is trivial.

Figures

Figures reproduced from arXiv: 2602.19861 by Asuka Shiga.

Figure 1
Figure 1. Figure 1: If #(F D(Q)/ℓF D(Q)) ≥ ℓ 2 and #X(ED/Q, Vis)[ℓ]/#X(ED/Q)[ℓ] < ℓ 2 , then X(ED/Q)[ℓ] ̸= 0. Proposition 4.8. Let ℓ be a prime, let E/Q be an elliptic curve, and let M ⊂ H1 (Qℓ , E)[ℓ] be a subgroup. Then #X(E/Q, M)[ℓ] #X(E/Q)[ℓ] ≤ #M. In particular, we obtain #X(E/Q, M)[ℓ] #X(E/Q)[ℓ] ≤ ℓ #E(Qℓ)[ℓ] [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
read the original abstract

Let $\ell$ be an odd prime. We study the visibility theorem for certain elliptic curves over $\mathbb{Q}$ with additive reduction at $\ell$, and deduce the existence of nontrivial $\ell$-torsion in $\Sha(E^D/\mathbb{Q})$ for suitable quadratic twists $E^D$. As an application for $\ell=3$, we exhibit pairs of non-isomorphic elliptic curves with the same BSD invariants, Kodaira symbols, and minimal discriminants, whose Tate--Shafarevich groups are isomorphic and have nontrivial $3$-primary parts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper studies the visibility theorem for elliptic curves over Q with additive reduction at an odd prime ℓ. It deduces the existence of nontrivial ℓ-torsion in Sha(E^D/Q) for suitable quadratic twists E^D of such curves. For ℓ=3, it exhibits pairs of non-isomorphic elliptic curves sharing the same BSD invariants, Kodaira symbols, and minimal discriminants, with isomorphic Tate-Shafarevich groups having nontrivial 3-primary parts.

Significance. If the central deduction holds, the work supplies explicit constructions of elliptic curves with visible nontrivial ℓ-torsion in Sha via quadratic twists, together with concrete ℓ=3 examples of curves that agree on all standard arithmetic invariants yet possess isomorphic Sha groups with 3-torsion. Such examples are useful for testing conjectures on the structure of Sha and for illustrating the limitations of local-global invariants in distinguishing elliptic curves.

major comments (1)
  1. [Visibility theorem and application to twists] The visibility theorem is formulated for curves with additive reduction at ℓ, yet the main deduction applies it to quadratic twists E^D where ℓ divides D. Quadratic twisting changes the Kodaira symbol and valuation v_ℓ(Δ) at ℓ (typically increasing it by 6v_ℓ(D)). The local condition ensuring that the image of the ℓ-torsion point under the connecting homomorphism lies in Sha rather than a larger Selmer group may fail after this change. The manuscript should explicitly verify, in the section containing the visibility theorem and its application to twists, that the required local conditions at ℓ remain satisfied for the chosen D.
minor comments (2)
  1. [Examples for ℓ=3] Clarify the precise choice of D in the examples (e.g., whether D is always divisible by ℓ) and state the resulting Kodaira symbols for both E and E^D at ℓ.
  2. [Application section] The abstract refers to 'pairs of non-isomorphic elliptic curves'; the manuscript should include a short table or explicit Weierstrass equations for at least one such pair to make the application concrete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for an explicit local verification when applying the visibility theorem to quadratic twists. We address the major comment below and will revise the manuscript to incorporate the requested check.

read point-by-point responses

  1. Referee: [Visibility theorem and application to twists] The visibility theorem is formulated for curves with additive reduction at ℓ, yet the main deduction applies it to quadratic twists E^D where ℓ divides D. Quadratic twisting changes the Kodaira symbol and valuation v_ℓ(Δ) at ℓ (typically increasing it by 6v_ℓ(D)). The local condition ensuring that the image of the ℓ-torsion point under the connecting homomorphism lies in Sha rather than a larger Selmer group may fail after this change. The manuscript should explicitly verify, in the section containing the visibility theorem and its application to twists, that the required local conditions at ℓ remain satisfied for the chosen D.

    Authors: We agree that an explicit verification of the local conditions at ℓ is required after twisting. In the revised manuscript we will add a short lemma immediately following the visibility theorem, computing the Kodaira symbol of E^D at ℓ and confirming that the image of the ℓ-torsion point under the connecting homomorphism remains inside the subgroup defining Sha. The argument uses the explicit local Galois-cohomology description for additive reduction together with the fact that our chosen D (with v_ℓ(D) positive and even or odd as needed) produce twists whose local invariants at ℓ still satisfy the visibility criterion. This addition will make the deduction fully rigorous without altering any of the stated theorems or examples. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper deduces nontrivial ℓ-torsion in Sha(E^D/Q) by applying the visibility theorem to elliptic curves with additive reduction at ℓ and their quadratic twists. No load-bearing step reduces by construction to its own inputs via self-definition, fitted parameters renamed as predictions, or a self-citation chain that renders the result tautological. The central claim rests on external visibility results and explicit twist constructions whose applicability is argued directly rather than assumed by renaming or fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms or invented entities; all such items are therefore recorded as empty.

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Works this paper leans on

25 extracted references · 25 canonical work pages · 3 internal anchors

  1. [1]

    Agashe and S

    A. Agashe and S. Biswas, Constructing non-trivial elements of the Shafarevich–Tate group of an abelian variety over a number field, J. Number Theory 133 (2013), no. 6, 1977–1990

    work page 2013

  2. [2]

    Agashe and W

    A. Agashe and W. Stein, Visibility of Shafarevich–Tate groups of abelian varieties, J. Number Theory 97 (2002), no. 1, 171–185

    work page 2002

  3. [3]

    Agashe and W

    A. Agashe and W. Stein, Visible evidence for the Birch and Swinnerton–Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Comp. 74 (2005), no. 249, 455–484. 16 ASUKA SHIGA

    work page 2005

  4. [4]

    A. J. Barrios, M. Brucal-Hallare, A. Deines, P. Harris, M. Roy, Prime isogenous discriminant ideal twins, arXiv:2402.19183(2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  5. [5]

    A.J Barrios, M. Roy, N. Sahajpal, D. Tallana, B. Tobin, H. Wiersema. Local data of elliptic curves under quadratic twist. Res. number theory 11, 75 (2025). https://doi.org/10.1007/s40993-025-00650-w

    work page doi:10.1007/s40993-025-00650-w 2025

  6. [6]

    Bhargava, Z

    M. Bhargava, Z. Klagsbrun, R. J. Lemke Oliver and A. Shnidman, Elements of given order in Tate–Shafarevich groups of abelian varieties in quadratic twist families, Algebra Number Theory 15 (2021), no. 3, 627–655

    work page 2021

  7. [7]

    Cassels, Arithmetic on curves of genus 1

    J.W.S. Cassels, Arithmetic on curves of genus 1. VII. The dual exact sequence. Journal für die reine und angewandte Mathematik 216: 150-158, 1964

    work page 1964

  8. [8]

    J. E. Cremona and N. Freitas, Global methods for the symplectic type of congruences between elliptic curves, Rev. Mat. Iberoam. 38 (2022), no. 1, 1–32, doi:10.4171/RMI/1269

    work page doi:10.4171/rmi/1269 2022

  9. [9]

    J. E. Cremona and B. Mazur, Visualizing elements in the Shafarevich–Tate group, Experiment. Math. 9 (2000), no. 1, 13–28

    work page 2000

  10. [10]

    Dokchitser and V

    T. Dokchitser and V. Dokchitser, On the Birch-Swinnerton-Dyer quotients modulo squares, Annals of Mathe- matics172(2010), no. 1, 567–596

    work page 2010

  11. [11]

    Fisher, Visualizing elements of order7in the Tate–Shafarevich group of an elliptic curve, LMS J

    T. Fisher, Visualizing elements of order7in the Tate–Shafarevich group of an elliptic curve, LMS J. Comput. Math. 19 (Special Issue A) (2016), 100–114

    work page 2016

  12. [12]

    Ramanujan Math

    Kęstutis Česnavičius, Selmer groups as flat cohomology groups, J. Ramanujan Math. Soc.31 (2016), no. 1, 31–61

    work page 2016

  13. [13]

    The LMFDB Collaboration. (2025). The L-functions and modular forms database.https://www.lmfdb.org

    work page 2025

  14. [14]

    Abelian varieties overp-adic ground fields

    A. Mattuck, “Abelian varieties overp-adic ground fields”, Ann. of Math. (2)62(1955), 92–119. MR Zbl

    work page 1955

  15. [15]

    Mazur and K

    B. Mazur and K. Rubin, Finding large Selmer rank via an arithmetic theory of local constants, Ann. of Math. (2) 166 (2007), no. 2, 579–612

    work page 2007

  16. [16]

    On p-torsion of p-adic elliptic curves with additive reduction

    R. Pannekoek, Onp-torsion ofp-adic elliptic curves with additive reduction, arXiv:1211.5833 (2013)

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  17. [17]

    How to Do a p-Descent on an Elliptic Curve,

    Edward F. Schaefer and Michael Stoll, “How to Do a p-Descent on an Elliptic Curve,” Transactions of the American Mathematical Society 356 (2004), no. 3, 1209–1231

    work page 2004

  18. [18]

    Shiga, Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants, arXiv:2507.18574 (2025)

    A. Shiga, Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants, arXiv:2507.18574 (2025)

    work page arXiv 2025

  19. [19]

    Behaviors of the Tate--Shafarevich group of elliptic curves under quadratic field extensions

    A. Shiga, Behaviors of the Tate–Shafarevich group of elliptic curves under quadratic field extensions, 2024, arXiv:2411.12316, to appear in Tokyo Journal of Mathematics

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  20. [20]

    Shnidman and A

    A. Shnidman and A. Weiss, Elements of prime order in Tate–Shafarevich groups of abelian varieties overQ, Forum Math. Sigma 10 (2022), e98, 1–10

    work page 2022

  21. [21]

    J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994

    work page 1994

  22. [22]

    Stein, Modular Forms, a Computational Approach (Graduate Studies in Mathematics, 79)

    W. Stein, Modular Forms, a Computational Approach (Graduate Studies in Mathematics, 79)

    work page

  23. [23]

    Stein and C

    W. Stein and C. Wuthrich, Algorithms for the arithmetic of elliptic curves using Iwasawa theory, Math. Comp. 82 (2013), no. 283, 1757–1792, doi:10.1090/S0025-5718-2012-02649-4

    work page doi:10.1090/s0025-5718-2012-02649-4 2013

  24. [24]

    Sutherland, A local-global principle for rational isogenies of prime degree, Journal de théorie des nombres de Bordeaux 24 (2012), no

    Andrew V. Sutherland, A local-global principle for rational isogenies of prime degree, Journal de théorie des nombres de Bordeaux 24 (2012), no. 2, 475–485, doi:10.5802/jtnb.807

    work page doi:10.5802/jtnb.807 2012

  25. [25]

    Wiese, Modular Galois Representations and Applications, lecture notes (4 lectures held at the Higher School of Economics in Moscow, 2–4 April 2013), Version of 6 April 2013

    G. Wiese, Modular Galois Representations and Applications, lecture notes (4 lectures held at the Higher School of Economics in Moscow, 2–4 April 2013), Version of 6 April 2013. Available athttps://math.uni.lu/wiese/ notes/2013-ModGalRepApp.pdf(accessed 2025-12-20). Mathematical Institute, Graduate School of Science, Tohoku University, 6-3 Aramaki Aza Aoba...

    work page 2013