Papers / Talks etc. | 明日の数学gallery

Paper (newest first)

1. Asuka Shiga, Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants, https://arxiv.org/abs/2507.18574

(2025)

Regarding the problem of whether elliptic curves can be identified by BSD invariants (i.e., whether BSD invariants are complete invariants), we proved that there exist infinitely many pairs that have the same BSD invariants, the same Kodaira symbols, and the same minimal discriminants, yet are non-isomorphic (moreover, they have different j-invariants and are not isomorphic over any field extension). It turns out that if we do not distinguish twists, there is remarkably only one such pair, and for one chosen pair, we found that infinitely many more examples can be found among twists with positive density.

The proof works by making the 2-parts of Sha simultaneously zero, thereby making the entire group structure of Sha completely coincide. For the simultaneous vanishing of 2-parts, there are two approaches: a method using Alexander Smith's recent density theorem, and a method that gives explicit infinite families via 2-descent for balanced isogenies.

2. Asuka Shiga, Behaviors of the Tate--Shafarevich group of elliptic curves under quadratic field extensions,

https://arxiv.org/abs/2411.12316 (2024) updated version

Commentary: It has long been known from Cassels' theory of Tamagawa ratios that Sha(E_D/Q)[2] can be made large (by increasing ramification), and it immediately follows that Sha(E/Q(√D))[2] becomes large in conjunction with this. However, focusing on the fact that what is truly becoming large is their ratio, we investigated the behavior of this ratio. This research can also be described as studying the 2-torsion subgroup version of Yu's formula. We also addressed the question of whether Sha(E/Q(√D))[2] and Sha(E_d/Q)[2] can be simultaneously reduced.

Talk (newest first)

2025 Academic Year

Ishigaki Island Algebraic Geometry Workshop, September 5-7, 2025 (scheduled), invited!
Y-RANT Young researchers in number theory, University of Nottingham, September 3 (scheduled)
On infinite families of non-isomorphic elliptic curves sharing BSD invariants, Sendai-Hiroshima Number Theory Meeting, July 10, 2025.

2024 Academic Year

On the behavior of Tate-Shafarevich groups of elliptic curves under quadratic extensions, Mathematical Society of Japan, Waseda University, March 18, 2025
On the behavior of elements of order $n$ ($n \geq 1$) in Tate-Shafarevich groups of elliptic curves under quadratic extensions, Kyushu Algebraic Number Theory 2025, Kyushu University, March 3, 2025
Relationship between quadratic extensions and twists of Tate-Shafarevich groups, Tsuda University Number Theory Workshop 2024, Tsuda University, November 23, 2024, invited!
Tate-Shafarevich group analogy of genus theory via Poitou-Tate duality and 2-descent, L-functions and Motives in Niseko, September 15-20, 2024 (poster presentation)
On behaviors of the 2-torsion subgroup of the Tate-Shafarevich group under quadratic number field extensions, 23rd Sendai-Hiroshima Number Theory Meeting, Tohoku University, July 12, 2024

2023 Academic Year

On the behavior of Tate-Shafarevich groups of elliptic curves under field extensions, 20th General Meeting of Young Researchers in Mathematics ~Crossroads of Mathematics~, Hokkaido University, March 4, 2024
On the behavior of Tate-Shafarevich groups of elliptic curves under field extensions, 3rd Budding of Algebra, Tokyo University of Science, February 28, 2024, invited!
On the behavior of Tate-Shafarevich groups of elliptic curves under quadratic extensions, 7th Mathematical Newcomers Seminar, Nagoya University, February 24, 2024

Internship

1. NTT Mathematical Foundations Center Summer Internship (August 13 - September 6, 2024)

awards

Master's Thesis Kawai Encouragement Award, March 2024