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

TL;DR

This work analyzes how the $2$-torsion in the Tate–Shafarevich group of an elliptic curve $E$ over a number field behaves under quadratic extensions and twists. By developing a local–global cohomology framework and introducing the finite cokernel $X$ of the restriction map, the authors connect $\nabla\Sha$ growth to the $2$-Selmer group via $X \cong \mathrm{Sel}^2(E/K)^*$. They then leverage Yu’s formula, trace–twist interactions, and $2$-descent to construct families of quadratic fields and twists with prescribed $2$-torsion behavior: both unbounded growth and vanishing results are obtained under finiteness assumptions, with explicit results for curves of the form $E:y^2=x^3+px$ and primes $p$. Notably, they show that for suitable $E$ there exist infinitely many $D$ with $\#\Sha(E/\mathbb{Q}(\sqrt{D}))[4]$ large while $\#\Sha(E_D/\mathbb{Q})[2]$ is simultaneously large, and, in many cases, $\Sha(E_D/\mathbb{Q})[2]=0$ for infinitely many $D$; however, certain congruence conditions (e.g., $p=257$) enforce non-vanishing of $\Sha(E/\mathbb{Q}(\sqrt{-D}))[2]$ for all $D$. The results illuminate the delicate balance between local contributions and global obstructions in the arithmetic of elliptic curves under quadratic twists and field extensions.

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 $p=257$.

Figures (2)

• Figure 1: A key diagram
• Figure 2: The gap between $\mathrm{Ker}H$ and $\Sha(E/L)[2]$

Theorems & Definitions (48)

• Theorem 1.1: Theorem \ref{['cor']}
• Theorem 1.2: Theorem \ref{['Zhe']}
• Proposition 1.3: Proposition \ref{['main theorem 3']}
• Theorem 3.1: cf. Neu, (8.6.10), Long Exact Sequence of Poitou--Tate
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Proposition 3.4
• proof