16 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…

Even-dimensionality conjecture. The group $\Sha(E)[2]$ has even dimension as an $\mathbb{F}_2$-vector space. This is expected to follow from the conjectural finiteness of $\Sha(E)$…

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…

Let $p$ be a prime, let $k$ be a fixed global field, and let $E$ vary over elliptic curves over $k$. Consider the short exact sequence … Using the natural discrete probability dist…

Let $k\geq 0$ be an integer and let $n=p_0p_1\cdots p_k$ be a product of distinct odd primes, with $p_i\equiv 1\pmod{8}$ for $1\leq i\leq k$. Let $m=n$ or $2n$ satisfy $m\equiv 5$,…

Birch and Swinnerton-Dyer predicted-order formula. The conjectural order of the Shafarevich–Tate group is

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

Let $C$ be a genus-one curve over ${\mathbb Q}$ with points everywhere locally, and let…

Let $E$ be an elliptic curve over ${\bf Q}$. Let $L_E(s)$ be its $L$-function, let $r$ be the order of vanishing of $L_E(s)$ at $s=1$, let $c_p(E)$ be the order of the arithmetic c…

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 $\text{\cyrr Sh}(E/K)$ denote its Shafarevich-Tate group. Shafarevich-Tate 2-parity conjecture. For every elli…

Higher-level visibility conjecture. Every element of the Shafarevich–Tate group can be explained by visibility at some higher level.

Let $f$ be the newform and $A_f$ the associated modular abelian variety. For an abelian variety $J$ equipped with a map $A_f^\vee\to J$, an element of…

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…