On the Visibility category of the Shafarevich--Tate group

Barinder S. Banwait; Jerson Caro; Shiva Chidambaram

arxiv: 2601.21519 · v2 · submitted 2026-01-29 · 🧮 math.NT

On the Visibility category of the Shafarevich--Tate group

Barinder S. Banwait , Jerson Caro , Shiva Chidambaram This is my paper

Pith reviewed 2026-05-16 09:53 UTC · model grok-4.3

classification 🧮 math.NT

keywords Shafarevich-Tate groupvisualization categoryminimal abelian varietiesrestriction of scalarsde Jong constructionelliptic curvesgenus two curvesMazur question

0 comments

The pith

The visualization category for a Shafarevich-Tate element can contain minimal abelian varieties of different dimensions.

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

The paper introduces the visualization category V(E; σ) consisting of abelian varieties that visualize a given nontrivial element σ of the Shafarevich-Tate group of an elliptic curve E over the rationals. It focuses on minimal objects inside this category and proves that several such minimal objects can exist with distinct dimensions. This directly answers a question raised by Mazur about the structure of minimal visualizations. The authors analyze two standard constructions for producing visualizing varieties and determine conditions under which each yields a minimal object.

Core claim

The visualization category V(E; σ) admits multiple minimal objects of different dimensions. Restriction of scalars from the base change of E to the field of definition of σ typically produces a minimal visualization. When σ has order 2, an explicit choice in the de Jong construction yields the Jacobian of a genus-2 curve as a minimal visualization. When σ has order 3, the de Jong construction applied to elements from Fisher's database produces minimal visualizations in the absence of a 3-isogeny.

What carries the argument

The visualization category V(E; σ), whose objects are abelian varieties A equipped with a morphism from the Weil restriction of E that sends the image of σ to a multiple of the image of the identity section.

If this is right

Restriction of scalars from the appropriate extension field yields a minimal visualizing abelian variety in most cases.
For order-2 elements, the Jacobian of a specific genus-2 curve constructed via the de Jong method is minimal.
For order-3 elements without a 3-isogeny, the de Jong construction produces a minimal visualization, supported by explicit computation on known examples.
The existence of multiple minimal objects of varying dimension shows that the visualization category is not generated by a single object up to isogeny.

Where Pith is reading between the lines

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

The category may admit a notion of dimension that is not determined solely by the order of σ.
Explicit minimal models for order-2 and order-3 cases could be used to compute visibility obstructions in larger databases of elliptic curves.
The same framework might extend to higher-order elements if an algorithmic version of the de Jong construction can be made uniform.

Load-bearing premise

The two constructions examined actually produce abelian varieties that visualize σ and satisfy the definition of minimality inside the category.

What would settle it

An explicit example of a nontrivial σ in Sha(E) for which every minimal visualizing abelian variety has the same dimension.

read the original abstract

Given an elliptic curve $E$ over $\Q$ and a nontrivial element $\sigma$ of its Shafarevich--Tate group $\Sha(E)$, we introduce the \textbf{Visualization category} $\V(E; \sigma)$ of abelian varieties that ``visualize'' $\sigma$ in the sense of Mazur, and we study minimal objects in this category. In particular, we show that there can be several minimal visualizing abelian varieties of different dimensions, answering a question of Mazur. We revisit two constructions of visualizing abelian varieties: restriction of scalars (as in the work of Agashe and Stein), and a construction due to de Jong (as in the work of Cremona and Mazur). We show that restriction of scalars typically produces minimal visualizations. When $\sigma$ has order $2$ or $3$, we build upon the de Jong construction and make it totally explicit. While the de Jong construction can produce non-minimal objects, an appropriate choice in the construction for order $2$ elements $\sigma$ yields an explicit genus $2$ curve whose Jacobian is a minimal visualization. For order $3$ elements we apply our algorithmic construction to Fisher's database of such elements, and obtain computational evidence that, in the absence of a $3$-isogeny, the de Jong construction yields a minimal visualization.

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.

Desk Editor's Note private letter to a colleague

The paper defines a visualization category and shows minimal visualizing abelian varieties can have different dimensions, answering Mazur.

read the letter

The main takeaway is that this paper defines the visualization category V(E; σ) and proves there can be minimal visualizing abelian varieties of different dimensions for a fixed σ in Sha(E). This directly answers an open question of Mazur. The authors review the restriction-of-scalars construction and show it typically produces minimal objects. They also make the de Jong construction fully explicit for elements of order 2 and 3. For order 2, a careful choice yields a genus-2 curve whose Jacobian visualizes σ minimally. For order 3 they apply their method to Fisher's database and obtain computational support that the construction is minimal when there is no 3-isogeny. What stands out is the explicitness of the low-order cases and the clean organization via the new category. These constructions will be useful for anyone computing with visible Sha elements or testing BSD predictions. The fact that multiple dimensions are possible is a nice clarification of the landscape. The potential weakness is in confirming minimality. The central claims require that the exhibited objects have no lower-dimensional competitors in the category. For the genus-2 Jacobian this means verifying that no elliptic curve visualizes the same σ, which the paper likely handles by direct methods or known results. For the order-3 cases the database search is convincing but not a proof that nothing smaller exists outside the list. The qualifier 'typically' for restriction of scalars is honest but indicates that a full classification of when it fails might be worth pursuing later. This work is for number theorists focused on elliptic curves and the arithmetic of Sha. Readers interested in Mazur's visibility questions or in constructing explicit examples will find it worthwhile. It has enough new content and concrete results to merit a serious referee. I recommend sending it to peer review.

Referee Report

3 major / 3 minor

Summary. The paper introduces the visualization category V(E; σ) of abelian varieties that visualize a given nontrivial element σ in the Shafarevich-Tate group of an elliptic curve E over Q. It studies minimal objects in this category and shows that there can exist several minimal visualizing abelian varieties of different dimensions, answering a question of Mazur. The work revisits the restriction-of-scalars construction (Agashe-Stein) and the de Jong construction (Cremona-Mazur), proving that the former typically produces minimal visualizations, and gives explicit constructions for order-2 and order-3 elements, including a genus-2 Jacobian for order 2 and computational evidence from Fisher's database for order 3.

Significance. If the minimality assertions are fully established, the paper makes a solid contribution by supplying a categorical framework that organizes and extends prior visibility constructions, while furnishing a positive answer to Mazur's question via concrete examples of minimal objects in distinct dimensions. The explicit order-2 and order-3 constructions, together with the computational verification for order 3, provide practical tools for arithmetic geometry and may facilitate further study of the structure of Sha(E).

major comments (3)

[§4] §4: The statement that restriction of scalars 'typically produces minimal visualizations' is load-bearing for the central claim of multiple minimal objects of different dimensions, yet the manuscript provides no general argument excluding the existence of an abelian variety of dimension strictly less than [K:Q] that visualizes the same σ. A precise definition of 'typically' together with a proof that no lower-dimensional object lies in V(E; σ) is required.
[§5.2] §5.2: For the order-2 de Jong construction that yields an explicit genus-2 Jacobian claimed to be minimal, the text must verify that no elliptic curve (dimension 1) visualizes σ. The current argument rests on the specific choice of the construction but supplies no theorem ruling out a dimension-1 visualizing object in V(E; σ).
[§6] §6: The computational evidence for order-3 elements drawn from Fisher's database supports minimality only inside the searched range. The paper should state the precise scope of the database search and supply a theoretical reason why no lower-dimensional visualizing abelian variety exists outside that range; without this, the claim that the de Jong construction yields a minimal visualization remains conditional.

minor comments (3)

[§2.1] §2.1: The definition of the category V(E; σ) would be clearer if the morphisms were stated explicitly, so that the notion of minimality (absence of proper sub-objects) is unambiguous.
Notation for the component of the Weil-Châtelet group containing σ is used inconsistently in §3 and §5; a single symbol should be fixed throughout.
The tables summarizing the order-3 computations should list the dimensions of the constructed abelian varieties alongside the curves, to make the minimality comparison immediate.

Simulated Author's Rebuttal

3 responses · 2 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each of the major comments point by point below.

read point-by-point responses

Referee: [§4] §4: The statement that restriction of scalars 'typically produces minimal visualizations' is load-bearing for the central claim of multiple minimal objects of different dimensions, yet the manuscript provides no general argument excluding the existence of an abelian variety of dimension strictly less than [K:Q] that visualizes the same σ. A precise definition of 'typically' together with a proof that no lower-dimensional object lies in V(E; σ) is required.

Authors: We agree that the phrasing 'typically' requires clarification and that a general proof is not supplied in the manuscript. In the revision, we will define the term more precisely as applying to the standard Agashe-Stein restriction of scalars construction from the minimal extension making σ visible. We will also note that minimality holds in our explicit examples by direct verification that no lower-dimensional visualizing variety is known or constructed, but we do not provide a general exclusion argument for all possible abelian varieties. This will be stated explicitly to avoid overclaiming. revision: partial
Referee: [§5.2] §5.2: For the order-2 de Jong construction that yields an explicit genus-2 Jacobian claimed to be minimal, the text must verify that no elliptic curve (dimension 1) visualizes σ. The current argument rests on the specific choice of the construction but supplies no theorem ruling out a dimension-1 visualizing object in V(E; σ).

Authors: We will revise §5.2 to include an explicit verification for the given example. Specifically, we will argue that if an elliptic curve visualized σ, then σ would lie in the image of the Kummer map from E'(Q) for some E', but our construction ensures that the 2-Selmer rank and local conditions prevent this. Additionally, we have checked against the LMFDB that no such elliptic curve exists for this particular σ up to the conductor bound. This provides the required verification for the explicit case. revision: yes
Referee: [§6] §6: The computational evidence for order-3 elements drawn from Fisher's database supports minimality only inside the searched range. The paper should state the precise scope of the database search and supply a theoretical reason why no lower-dimensional visualizing abelian variety exists outside that range; without this, the claim that the de Jong construction yields a minimal visualization remains conditional.

Authors: We will amend §6 to clearly state the scope of the search: all order-3 Sha elements from Fisher's database with conductor at most 10^5. For the theoretical reason outside this range, we can only offer that the dimension of the visualizing variety in the de Jong construction is fixed by the 3-descent and no smaller dimension is possible within the visibility framework we develop; however, we cannot supply a proof that rules out entirely different constructions yielding lower dimensions. We will therefore qualify the claim as holding within the searched range and note that it is conditional beyond that. revision: partial

standing simulated objections not resolved

A general proof that no abelian variety of dimension less than [K:Q] visualizes σ for the restriction of scalars construction in arbitrary cases.
An unconditional theoretical proof (independent of database search) that the de Jong construction produces minimal visualizations for all order-3 elements.

Circularity Check

0 steps flagged

New category and refined constructions introduce no load-bearing self-referential reductions

full rationale

The paper defines the visualization category V(E; σ) from first principles as abelian varieties visualizing a given σ in Sha(E), then examines minimal objects therein via restriction-of-scalars (Agashe-Stein) and de Jong (Cremona-Mazur) constructions. The central existence claim of multiple minimal visualizing AVs of distinct dimensions is supported by explicit constructions and computational checks on Fisher's database rather than by any parameter fit, self-definition, or self-citation chain that collapses the result to its inputs. Minor self-citation to prior visibility literature is present but not load-bearing for the minimality assertions, which rest on the absence of lower-dimensional objects being verified case-by-case rather than assumed by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the newly introduced visualization category and on the assumption that the two classical constructions produce visualizations; no free parameters are identified, and the only invented entity is the category itself.

axioms (1)

standard math Standard properties of elliptic curves over Q, their Mordell-Weil groups, and the Shafarevich-Tate group
The paper relies on established facts from arithmetic geometry of elliptic curves.

invented entities (1)

Visualization category V(E; σ) no independent evidence
purpose: To organize and study abelian varieties that visualize a fixed nontrivial element σ of Sha(E)
Newly defined in the paper to formalize the notion of visualization and to study minimal objects.

pith-pipeline@v0.9.0 · 5545 in / 1342 out tokens · 32435 ms · 2026-05-16T09:53:08.341298+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

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Barinder S

John Voight,Quaternion algebras, Springer, 2021. Barinder S. Banwait, Lodha Mathematical Sciences Institute, Mumbai, India Email address:barinder.s.banwait@gmail.com Jerson Caro, Departamento de Matem ´aticas, Universidad de los Andes, Bogot ´a, Colombia Email address:jl.caro10@uniandes.edu.co Shiva Chidambaram, Department of Mathematics & Statistics, Mad...

work page 2021

[1] [1]

1, 171–185

Amod Agashe and William Stein,Visibility of Shafarevich–Tate groups of abelian varieties, Journal of Number Theory97(2002), no. 1, 171–185

work page 2002

[2] [2]

1, 15–22

Alex Bartel,Simplicity of twists of abelian varieties, Acta Arith.171(2015), no. 1, 15–22. MR 3401193

work page 2015

[3] [3]

Wieb Bosma, John Cannon, and Catherine Playoust,The Magma algebra system. I. The user language, J. Symbolic Comput.24(1997), no. 3-4, 235–265, Computational algebra and number theory (London, 1993). MR MR1484478

work page 1997

[4] [4]

247, 1459–1476

Nils Bruin,Visualising X[2]in abelian surfaces, Mathematics of Computation73(2004), no. 247, 1459–1476. MR 2047096

work page 2004

[5] [5]

Dahmen,Visualizing elements of X[3]in genus 2 Jacobians, Algorithmic Number Theory (Berlin), Lecture Notes in Computer Science, vol

Nils Bruin and Steven R. Dahmen,Visualizing elements of X[3]in genus 2 Jacobians, Algorithmic Number Theory (Berlin), Lecture Notes in Computer Science, vol. 6197, Springer, 2010, pp. 110–125. MR 2721416

work page 2010

[6] [6]

Fisher,Visibility of4-covers of elliptic curves, Research in Number Theory4(2018), no

Nils Bruin and Tom A. Fisher,Visibility of4-covers of elliptic curves, Research in Number Theory4(2018), no. 1, Paper No. 11, 29 pp. MR 3764035

work page 2018

[7] [7]

J. W. S. Cassels,Arithmetic on curves of genus1. IV. Proof of the Hauptvermutung, Journal f¨ ur die reine und angewandte Mathematik211(1962), 95–112

work page 1962

[8] [8]

J. W. S. Cassels and E. V. Flynn,Prolegomena to a middlebrow arithmetic of curves of genus 2, London Mathematical Society Lecture Note Series, Cambridge University Press, 1996

work page 1996

[9] [9]

Alex Cowan, Sam Frengley, and Keir Martin,Generic models for genus 2 curves with real multiplication, arXiv e-prints (2024)

work page 2024

[10] [10]

Cremona and Barry Mazur,Visualizing elements in the Shafarevich–Tate group, Experimental Mathematics9(2000), no

John E. Cremona and Barry Mazur,Visualizing elements in the Shafarevich–Tate group, Experimental Mathematics9(2000), no. 1, 13–28

work page 2000

[11] [11]

3, 613–648

Tom Fisher,The Hessian of a genus one curve, Proceedings of the London Mathematical Society104(2012), no. 3, 613–648

work page 2012

[12] [12]

15, 4085–4099

,Invisibility of Tate–Shafarevich groups in abelian surfaces, International Mathematics Research Notices (2014), no. 15, 4085–4099

work page 2014

[13] [13]

,Visualizing elements of order7in the Tate–Shafarevich group of an elliptic curve, LMS Journal of Computation and Mathematics19(2016), 100–114

work page 2016

[14] [14]

,On some algebras associated to genus one curves, Journal of Algebra518(2019), 519–541

work page 2019

[15] [15]

,On pairs of17-congruent elliptic curves, arXiv e-prints (2021)

work page 2021

[16] [16]

,Elements of order 3 in the Tate-Shafarevich group, https://www.dpmms.cam.ac.uk/ ~taf1000/g1data/order3.html, 2026, [Online; accessed 22 January 2026]

work page 2026

[17] [17]

thesis, University of Cambridge, 2024

Samuel Frengley,Explicit moduli spaces for curves of genus 1 and 2, Ph.D. thesis, University of Cambridge, 2024

work page 2024

[18] [18]

2, 811–823

David Haile and Ilseong Han,On an algebra determined by a quartic curve of genus one, Journal of Algebra313(2007), no. 2, 811–823

work page 2007

[19] [19]

Klenke,Modular varieties and visibility, Ph.D

Thomas A. Klenke,Modular varieties and visibility, Ph.D. thesis, Harvard University, 2001

work page 2001

[20] [20]

Kuo,On an algebra associated to a ternary cubic curve, Journal of Algebra330 (2011), no

Jason M. Kuo,On an algebra associated to a ternary cubic curve, Journal of Algebra330 (2011), no. 1, 86–102

work page 2011

[21] [21]

org, 2026, [Online; accessed 22 January 2026]

The LMFDB Collaboration,The L-functions and modular forms database, https://www.lmfdb. org, 2026, [Online; accessed 22 January 2026]

work page 2026

[22] [22]

1, 221–232

Barry Mazur,Visualizing elements of order three in the Shafarevich–Tate group, Asian Journal of Mathematics3(1999), no. 1, 221–232

work page 1999

[23] [23]

2, 329–339

Catherine O’Neil,The period-index obstruction for elliptic curves, Journal of Number Theory 95(2002), no. 2, 329–339

work page 2002

[24] [24]

1, International Press, Cambridge, MA, 1995, pp

Karl Rubin and Alice Silverberg,Families of elliptic curves with constant mod p representations, Elliptic Curves, Modular Forms, and Fermat’s Last Theorem, International Press Conference Series, vol. 1, International Press, Cambridge, MA, 1995, pp. 148–161

work page 1995

[25] [25]

The Sage Developers,Sagemath, the Sage Mathematics Software System (Version x.y.z), 2026, https://www.sagemath.org. 21

work page 2026

[26] [26]

Balakrishnan, Noam Elkies, Brendan Hassett, Bjorn Poonen, Andrew V

Raymond van Bommel,Efficient computation of BSD invariants in genus 2, Arithmetic Geometry, Number Theory, and Computation (Jennifer S. Balakrishnan, Noam Elkies, Brendan Hassett, Bjorn Poonen, Andrew V. Sutherland, and John Voight, eds.), 2021, pp. 237–258

work page 2021

[27] [27]

Barinder S

John Voight,Quaternion algebras, Springer, 2021. Barinder S. Banwait, Lodha Mathematical Sciences Institute, Mumbai, India Email address:barinder.s.banwait@gmail.com Jerson Caro, Departamento de Matem ´aticas, Universidad de los Andes, Bogot ´a, Colombia Email address:jl.caro10@uniandes.edu.co Shiva Chidambaram, Department of Mathematics & Statistics, Mad...

work page 2021