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arxiv: 2411.12316 · v3 · submitted 2024-11-19 · 🧮 math.NT

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

Asuka Shiga This is my paper

Pith reviewed 2026-05-23 17:47 UTC · model grok-4.3

classification 🧮 math.NT
keywords Tate-Shafarevich groupelliptic curvequadratic extensionquadratic twistrestriction map2-torsioncokernel
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The pith

The ratio of Sha[4] over a quadratic extension to Sha[2] of the twist, together with the latter order, can grow arbitrarily large for some elliptic curves without assuming finiteness of Sha.

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

The paper examines the Tate-Shafarevich group of an elliptic curve E over Q when base-changed to quadratic extensions Q(sqrt(D)). By analyzing the cokernel of the restriction map, it shows that for suitable E the ratio of the order of the 4-torsion in Sha(E/Q(sqrt(D))) to the order of the 2-torsion in Sha of the quadratic twist E_D, along with that 2-torsion order itself, can become arbitrarily large at the same time. This holds without any finiteness assumption on Sha. For the concrete family of curves given by y^2 = x^3 + p x where p is an odd prime congruent to 1 modulo 4, the assumption that the relevant Sha groups are finite yields the stronger statement that Sha(E/Q(sqrt(D)))[2] has order at most 4 and Sha(E_D/Q)[2] vanishes, for infinitely many square-free D with -D prime. When p equals 257 the 2-torsion in Sha(E/Q(sqrt(-D))) is always nontrivial.

Core claim

By analyzing the cokernel of the restriction map, the ratio #Sha(E/Q(sqrt(D)))[4] / #Sha(E_D/Q)[2] and #Sha(E_D/Q)[2] can grow arbitrarily large simultaneously under some conditions on E/Q without assuming finiteness of Sha; for E : y^2 = x^3 + p x with p odd prime ≡1 mod 4, #Sha(E/Q(sqrt(D)))[2] ≤4 and Sha(E_D/Q)[2]=0 for infinitely many D with -D prime, assuming finiteness. Additionally, Sha(E/Q(sqrt(-D)))[2] ≠0 for all such D when p=257.

What carries the argument

cokernel of the restriction map from Sha(E/Q) to Sha(E/Q(sqrt(D)))

If this is right

  • The 2-primary parts of Sha(E/Q(sqrt(D))) and Sha(E_D/Q) can be made arbitrarily large simultaneously by choice of D for appropriate E.
  • For the family y^2 = x^3 + p x the 2-torsion in Sha of the twist vanishes for infinitely many D with -D prime.
  • When p=257 the 2-torsion in Sha(E/Q(sqrt(-D))) is nontrivial for every such D.

Where Pith is reading between the lines

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

  • The growth result without finiteness may allow construction of elliptic curves over quadratic fields whose Sha[2] orders are arbitrarily large.
  • The boundedness result for the specific family provides an explicit infinite set of quadratic twists with controlled Sha[2].
  • The contrast between unbounded growth in general and boundedness for the family highlights the role of the curve's Weierstrass equation in controlling the cokernel.

Load-bearing premise

Finiteness of the relevant Tate-Shafarevich groups is assumed when proving the bound of 4 and the vanishing for the specific family of curves y^2 = x^3 + p x.

What would settle it

An explicit D with -D prime, for the family with p odd prime ≡1 mod 4, such that either #Sha(E/Q(sqrt(D)))[2] >4 or #Sha(E_D/Q)[2] >0 under the finiteness assumption; or conditions on E where the simultaneous growth of the ratio and the order remains bounded.

Figures

Figures reproduced from arXiv: 2411.12316 by Asuka Shiga.

Figure 1
Figure 1. Figure 1: A key diagram Notably, by Theorem 3.3, X ∼= Sel2 (E/K) holds. Lemma 4.7. — #KerH = #X(E/K)[2]#L v∈MK H1 (Gal(Lw/Kv), E(Lw)) #j(CokerF)#H1(Gal(L/K), E(L)) . Proof. — By applying the snake lemma, 0 → KerF → X(E/K)[2] → KerH → CokerF → j(CokerF) → 0. We obtain #KerH = #X(E/K)[2]#CokerF #j(CokerF)#KerF . The left vertical exact sequence implies #CokerF #kerF = # L v∈MK H1 (Gal(Lw/Kv), E(Lw)) #H1(Gal(L/K), E(L)… view at source ↗
Figure 2
Figure 2. Figure 2: The gap between KerH and X(E/L)[2] Lemma 4.8. — #X(E/L)[2] #KerH ≥ #tr(X(E/L)[2]). Proof. — Since KerH ⊂ X(E/L)[2]Gal(L/K) ⊂ X(E/L)[2] and X(E/L)[2] X(E/L)[2]Gal(L/K) ∼= (σ − 1)X(E/L)[2] = tr(X(E/L)[2]), we obtain #X(E/L)[2] ≥ #tr(X(E/L)[2])#KerH. By combining Lemma 4.7 and Lemma 4.8 and Proposition 4.6, we obtain #X(E/L)[2] ≥ #tr(X(E/L)[2])#KerH (by Lemma 4.8) ≥ #X(ED/K)[2] 4 rank(E/K)#E(K)[2]2#2X(E/L)[4]… view at source ↗
read the original abstract

Let $E/\mathbb{Q}$ be an elliptic curve. We study the behavior of the Tate--Shafarevich group of $E$ under quadratic extensions $\mathbb{Q}(\sqrt{D})/\mathbb{Q}$. By analyzing the cokernel of the restriction map, without assuming the finiteness of the Tate--Shafarevich group, we prove that the ratio $\frac{\#\Sha(E/\mathbb{Q}(\sqrt{D}))[4]}{\#\Sha(E_D/\mathbb{Q})[2]}$ and $\#\Sha(E_D/\mathbb{Q})[2]$ can, under some conditions on $E/\mathbb{Q}$, grow arbitrarily large simultaneously, where $E_D$ denotes the quadratic twist of $E$ by $D$. For elliptic curves of the form $E : y^2 = x^3 + px$ with $p\equiv 1 \bmod 4$ being an odd prime, assuming the finiteness of the relevant Tate--Shafarevich groups, we prove that $\#\Sha(E/\mathbb{Q}(\sqrt{D}))[2] \leq 4$ and $\Sha(E_D/\mathbb{Q})[2] = 0$ for infinitely many square-free integers $D$ with $-D$ being a prime number. Additionally, $\Sha(E/\mathbb{Q}(\sqrt{-D}))[2]\neq 0$ for all $D$ when $p=257$.

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

2 major / 2 minor

Summary. The paper examines the Tate-Shafarevich group Sha of an elliptic curve E/Q under quadratic extensions Q(sqrt(D))/Q. It proves unconditionally, via cokernel analysis of restriction maps, that under stated conditions on E the ratio #Sha(E/Q(sqrt(D)))[4]/#Sha(E_D/Q)[2] and #Sha(E_D/Q)[2] can grow arbitrarily large simultaneously. For the specific family E: y^2 = x^3 + p x with p an odd prime ≡1 mod 4, assuming finiteness of the relevant Sha groups, it proves #Sha(E/Q(sqrt(D)))[2] ≤4 and Sha(E_D/Q)[2]=0 for infinitely many square-free D with -D prime; additionally, for p=257, Sha(E/Q(sqrt(-D)))[2] ≠0 for all such D.

Significance. If the results hold, the unconditional growth statements on the ratio and on #Sha(E_D/Q)[2] via explicit cokernel analysis constitute a concrete contribution to the study of Sha under base change and twisting, without relying on the BSD conjecture or finiteness. The separation of the unconditional part from the conditional bounds for the explicit Weierstrass family, together with the arithmetic conditions on D, strengthens the manuscript. The explicit verification for p=257 adds a falsifiable prediction.

major comments (2)
  1. [§3] §3 (or the section containing the cokernel computation): the claim that the ratio and #Sha(E_D/Q)[2] grow arbitrarily large simultaneously rests on the surjectivity or dimension of the cokernel of the restriction map res: Sha(E/Q)[4] → Sha(E/Q(sqrt(D)))[4]; the manuscript must exhibit an explicit infinite family of D (or a density statement) where the cokernel dimension increases without bound while the denominator remains controlled, otherwise the simultaneous growth is only shown to be possible rather than realized.
  2. [The section on the explicit family] The conditional result for the family y^2 = x^3 + p x (p ≡1 mod 4 prime): the bound #Sha(E/Q(sqrt(D)))[2] ≤4 and the vanishing Sha(E_D/Q)[2]=0 are stated under the finiteness assumption; the proof must verify that the 2-Selmer rank computations or the explicit 2-descent remain valid uniformly for the infinitely many D with -D prime, and that no additional local conditions at primes dividing the conductor are overlooked when D varies.
minor comments (2)
  1. The notation E_D for the quadratic twist should be defined at first use and kept consistent with the standard convention (twist by D or by the fundamental discriminant).
  2. In the statement for p=257, clarify whether the non-vanishing Sha(E/Q(sqrt(-D)))[2] ≠0 holds for the same infinite set of D or for all square-free D with -D prime.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive comments. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§3] §3 (or the section containing the cokernel computation): the claim that the ratio and #Sha(E_D/Q)[2] grow arbitrarily large simultaneously rests on the surjectivity or dimension of the cokernel of the restriction map res: Sha(E/Q)[4] → Sha(E/Q(sqrt(D)))[4]; the manuscript must exhibit an explicit infinite family of D (or a density statement) where the cokernel dimension increases without bound while the denominator remains controlled, otherwise the simultaneous growth is only shown to be possible rather than realized.

    Authors: The cokernel analysis in §3 is carried out in terms of the images of local restriction maps at primes dividing D. By taking D to be a product of k distinct primes, each satisfying a fixed set of congruence conditions that ensure the local images contribute independent elements to the cokernel, the cokernel dimension grows linearly with k. The same choice of primes forces #Sha(E_D/Q)[2] to grow at least linearly with k via the explicit description of the 2-Selmer group of the twist. We will add an explicit infinite family realizing this simultaneous growth in the revised manuscript. revision: yes

  2. Referee: [The section on the explicit family] The conditional result for the family y^2 = x^3 + p x (p ≡1 mod 4 prime): the bound #Sha(E/Q(sqrt(D)))[2] ≤4 and the vanishing Sha(E_D/Q)[2]=0 are stated under the finiteness assumption; the proof must verify that the 2-Selmer rank computations or the explicit 2-descent remain valid uniformly for the infinitely many D with -D prime, and that no additional local conditions at primes dividing the conductor are overlooked when D varies.

    Authors: The 2-descent for curves in this family is performed on the model y^2 = x^3 + p x and depends only on the local solubility conditions at the fixed primes 2 and p. The hypothesis that -D is prime ensures that D is coprime to the conductor of E and introduces no new local conditions at those primes; the local images in the Selmer groups are therefore identical for every such D. We will insert a short paragraph making this uniformity explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper separates an unconditional result on growth of the ratio #Sha(E/Q(sqrt(D)))[4]/#Sha(E_D/Q)[2] and #Sha(E_D/Q)[2] via cokernel analysis of the restriction map (no finiteness assumed) from a conditional result for the family y^2 = x^3 + p x that explicitly assumes finiteness of Sha to obtain the bound #Sha(E/Q(sqrt(D)))[2] <=4 and Sha(E_D/Q)[2]=0 for infinitely many D. Both rest on standard Galois-cohomology arguments with no self-definitional reductions, no fitted inputs renamed as predictions, and no load-bearing self-citations; the derivation chain is independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard background from elliptic curve theory and Galois cohomology; the only explicit extra assumption is finiteness of Sha for the boundedness statements.

axioms (2)
  • standard math Standard properties of the restriction map on Galois cohomology of elliptic curves and its cokernel
    Invoked to study Sha without assuming finiteness for the growth result.
  • domain assumption Finiteness of Sha(E/Q(sqrt(D))) and Sha(E_D/Q) for the family y^2 = x^3 + p x
    Explicitly assumed to obtain the bound #Sha[2] ≤4 and vanishing of Sha(E_D/Q)[2].

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    math.NT 2026-02 unverdicted novelty 5.0

    Visibility theorems imply nontrivial ℓ-torsion in Sha of quadratic twists of elliptic curves with additive reduction at ℓ; for ℓ=3 this yields pairs of curves with identical BSD data and Kodaira symbols but isomorphic...

Reference graph

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23 extracted references · 23 canonical work pages · cited by 1 Pith paper

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