9 problems

Let $K$ be a number field and let $\text{Ш}^{\operatorname{BK}}(K,\mathcal{M})=\bigoplus_p\text{Ш}_p^{\operatorname{BK}}(K,\mathcal{M})$ be the Bloch–Kato Shafarevich–Tate group of…

Let $E$ be an elliptic curve over a number field $K$, and let $\Sha(E/K)$ denote its Shafarevich–Tate group. Shafarevich–Tate finiteness conjecture. The group $\Sha(E/K)$ is finite…

Degenerate twistor deformation conjecture. There exists a degenerate twistor deformation of $\pi$ admitting a holomorphic section. This conjecture would remove the torsion-freeness…

Agashe's conjecture. Up to a power of $2$, the square of the order of $E^{-D}(\mathbb{Q})$ divides

Fix a global field $k$ and $r \in {\mathbb Z}_{\ge 0}$ such that ${\mathscr E}_r=\{E\in{\mathscr E}:\operatorname{rk}E(k)=r\}$ is infinite. For each finite abelian $p$-group $G$, c…

The quadratic-twist divisibility conjecture. The quantity $|E_{-D}(\mathbb{Q})|^2$ divides

Abundance conjecture for Shafarevich–Tate torsion. There exists an elliptic curve $E_{/k_0}$ such that…

Let $K$ be a number field and $E$ an elliptic curve over $K$, and let \cyrr Sh$(E/K)$ denote its Shafarevich-Tate group. Shafarevich-Tate 2-parity conjecture. For every elli…

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $r_{\rm an}(E/\mathbb{Q})$ be the order of vanishing of $L(E/\mathbb{Q},s)$ at $s=1$. For a prime $p$, let…