Papers

(in reverse chronological order)

1. Asuka Shiga, “Nontrivial Torsion in the Tate–Shafarevich Group via Visibility and Twists,” arXiv:2602.19861 (2026).

This paper investigates, using the theory of visibility, whether there exist pairs of non-isomorphic elliptic curves that share the same BSD invariants and have isomorphic nontrivial Tate–Shafarevich groups.

A particular difficulty in applying visibility theory has traditionally been the treatment of additive reduction. By relaxing the local conditions defining the Tate–Shafarevich group, I prove that visibility can still be applied when
[
E(\mathbb{Q}_{\ell})[\ell]=0,
]
a condition that can always be achieved after a suitable twist.

Groups obtained by modifying finitely many local conditions often play a key role. Examples include the fine Tate–Shafarevich group, for which the local conditions are strengthened, and the relaxed version introduced in this paper. It may also be interesting to construct visible elements in the fine Tate–Shafarevich group.

The visibility mechanism developed here provides sufficient conditions rather than necessary ones. When (E(\mathbb{Q}_{\ell})[\ell]\neq 0), a more delicate analysis of the composite map appearing in Remark 4.3 will likely be required.

2. Asuka Shiga, “Infinitely Many Pairs of Non-isomorphic Elliptic Curves Sharing the Same BSD Invariants,”

https://arxiv.org/abs/2507.18574
(2025)

Overview. This paper addresses the question of whether an elliptic curve is determined by its BSD invariants—that is, whether the BSD invariants form a complete set of invariants.

I prove that there exist infinitely many pairs of non-isomorphic elliptic curves having the same BSD invariants, the same Kodaira symbols, and the same minimal discriminant. Moreover, the two curves in each pair have different (j)-invariants and are therefore not isomorphic over any field extension.

Among elliptic curves satisfying
[
E(\mathbb{Q})\cong \mathbb{Z}/2\mathbb{Z},
]
there is, up to quadratic twist, only one such pair. Starting from this pair, however, a positive-density family of quadratic twists produces infinitely many further examples.

The proof makes the entire group structures of the Tate–Shafarevich groups coincide by forcing their (2)-primary parts to vanish simultaneously. There are two approaches to obtaining this simultaneous vanishing:

1. an argument using Alexander Smith’s recent density theorem; and

2. a (2)-descent argument that gives an explicit infinite family for balanced isogenies.

3. Asuka Shiga, “Behaviors of the Tate–Shafarevich Group of Elliptic Curves under Quadratic Field Extensions,” Tokyo Journal of Mathematics 49 (2026), no. 1, 199–224. DOI: 10.3836/tjm/1502179452. 

 arXiv link: https://arxiv.org/abs/2411.12316

​

Overview. It has long been known from Cassels’s theory of Tamagawa ratios that the group
[
\Sha(E_D/\mathbb{Q})[2]
]
can be made arbitrarily large by increasing the amount of ramification. This immediately implies a corresponding growth of
[
\Sha(E/\mathbb{Q}(\sqrt{D}))[2].
]

The main point of this paper is that the quantity that truly grows is the ratio between these groups. The paper studies the behavior of this ratio and may also be viewed as an investigation of the (2)-torsion analogue of Yu’s formula.

I also consider whether
[
\Sha(E/\mathbb{Q}(\sqrt{D}))[2]
\quad\text{and}\quad
\Sha(E_D/\mathbb{Q})[2]
]
can be made to decrease simultaneously.

Presentations

(in reverse chronological order)

Academic Year 2025

• Kyushu Algebraic Number Theory, March 2
  “Infinitely Many Pairs of Non-isomorphic Elliptic Curves Sharing the Same BSD Invariants.”

• Research on Calabi–Yau Varieties, Nagoya University, March 9
  “Infinitely Many Pairs of Non-isomorphic Elliptic Curves Sharing the Same BSD Invariants and Visibility in the Tate–Shafarevich Group.”

• Joint Number Theory Seminar, Kyoto University, January 16 (scheduled)
  “Infinitely Many Pairs of Non-isomorphic Elliptic Curves Sharing the Same BSD Invariants and Visibility in the Tate–Shafarevich Group.”
  Invited talk.

• Mini-Workshop on Arithmetic Geometry in Sendai, Tohoku University, December 22–24
  “Infinitely Many Pairs of Non-isomorphic Elliptic Curves Sharing the Same BSD Invariants and the Visibility Theorem.”
  Invited talk.

• Number Theory and (p)-adic Methods, University of Toyama, November 23–24, 2025
  “Infinitely Many Pairs of Non-isomorphic Elliptic Curves Sharing the Same BSD Invariants and the Visibility Theorem.”
  Poster presentation.

• Summer School on Number Theory, September 8–12
  “Infinitely Many Pairs of Non-isomorphic Elliptic Curves Sharing the Same BSD Invariants.”
  Poster presentation.

• Ishigaki Island Workshop on Algebraic Geometry, September 7, 2025
  “Infinitely Many Pairs of Non-isomorphic Elliptic Curves Sharing the Same BSD Invariants.”
  Invited talk.

• Y-RANT: Young Researchers in Number Theory, University of Nottingham, United Kingdom, September 3
  “Infinitely Many Pairs of Non-isomorphic Elliptic Curves Sharing the Same BSD Invariants.”

• Sendai–Hiroshima Workshop on Number Theory, July 10, 2025
  “Infinitely Many Pairs of Non-isomorphic Elliptic Curves Sharing the Same BSD Invariants.”

Academic Year 2024

• Mathematical Society of Japan, Waseda University, March 18, 2025
  “Behavior of the Tate–Shafarevich Group of an Elliptic Curve under Quadratic Extensions.”

• Kyushu Algebraic Number Theory 2025, Kyushu University, March 3, 2025
  “Behavior under Quadratic Extensions of Elements of Order (n\geq 1) in the Tate–Shafarevich Group of an Elliptic Curve.”

• Tsuda University Number Theory Workshop 2024, Tsuda University, November 23, 2024
  “The Relationship between Quadratic Extensions and Twists of the Tate–Shafarevich Group.”
  Invited talk.

• L-functions and Motives in Niseko, September 15–20, 2024
  “A Genus-Theory Analogue for the Tate–Shafarevich Group via Poitou–Tate Duality and (2)-Descent.”
  Poster presentation.

• 23rd Sendai–Hiroshima Workshop on Number Theory, Tohoku University, July 12, 2024
  “On the Behavior of the (2)-torsion Subgroup of the Tate–Shafarevich Group under Quadratic Number Field Extensions.”

Academic Year 2023

• 20th General Mathematics Workshop for Young Researchers: Crossroads of Mathematics, Hokkaido University, March 4, 2024
  “Behavior of the Tate–Shafarevich Group of an Elliptic Curve under Field Extensions.”

• Third Workshop on Emerging Algebra, Tokyo University of Science, February 28, 2024
  “Behavior of the Tate–Shafarevich Group of an Elliptic Curve under Field Extensions.”
  Invited talk.

• 7th Mathematical Sciences Newcomers Seminar, Nagoya University, February 24, 2024
  “Behavior of the Tate–Shafarevich Group of an Elliptic Curve under Quadratic Extensions.”

Internship

NTT Institute for Fundamental Mathematics Summer Internship

August 13–September 6, 2024

Project: A Computational Approach to Determining Rational Points on Curves

The project focused on the Mordell–Weil ranks and Tate–Shafarevich groups of elliptic curves with trivial torsion subgroup.

Final presentation: September 6, 2024

Awards

• Kawai Encouragement Award for Master’s Thesis, March 2024

Academic Service

• Seminar Chair, Tohoku University Number Theory Seminar, July 28
  Invited lecture by Alexandros Konstantinou:
  “The Order of the Tate–Shafarevich Group Modulo Squares.”