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

TL;DR

This paper investigates how the Tate--Shafarevich group of elliptic curves behaves under quadratic field extensions, revealing conditions under which certain parts of these groups can grow arbitrarily large or remain bounded.

Contribution

It provides new insights into the growth patterns of Tate--Shafarevich groups under quadratic extensions without assuming their finiteness, and establishes bounds for specific elliptic curves.

Findings

The ratio of certain Tate--Shafarevich group parts can grow arbitrarily large under some conditions.

For specific elliptic curves, the 2-part of the Tate--Shafarevich group is bounded by 4.

For some primes, the 2-part of the Tate--Shafarevich group is zero for infinitely many D.

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 $p=257$.