arXiv subjects

Asuka Shiga

Publications and source records attributed to Asuka Shiga.

3 recordsLinked to original sources

arXiv2026

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

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.

math.NT↗

arXiv2025

Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants

Let $E/\mathbb{Q}$ be an elliptic curve. The Birch and Swinnerton--Dyer (BSD) conjecture relates the leading coefficient of the Taylor expansion of the $L$-function of $E/\mathbb{Q}$ at $s=1$ to arithmetic invariants of $E$, such as its Mordell--Weil group, its Tate--Shafarevich group, its Tamagawa numbers, its regulator, and its real period. We call two non-isomorphic elliptic curves over $\mathbb{Q}$ BSD twins if they have the same $L$-function and the same arithmetic data underlying the BSD invariants appearing in the BSD conjecture, with the Mordell--Weil group and the Tate--Shafarevich group compared as groups. We exhibit a family of BSD twins for which the corresponding pairs of $j$-invariants are pairwise distinct. We further prove that, even after imposing equality of the Kodaira symbols at every prime and equality of the minimal discriminants, infinitely many BSD twins still exist.

math.NT↗

arXiv2024

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

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 $p=257$.

math.NT↗