16 problems

Visibility-dimension claim. This element is not visible in any abelian threefold; in particular, its visibility dimension is $4$.

Birch and Swinnerton-Dyer conjecture. One has

Rank-one joint distribution conjecture. As $d$ ranges over $\mathcal{F}$, the joint distribution of $\log|L'(\frac12,E_d)|$ and $\log(|\Sha(E_d)|R(E_d)/\sqrt{|d|})$ is asymptotical…

Jetchev–Stein conjecture. Every class in $\Sh(E/\mathbf{Q})$ is modular.

Let $J_i\to\mathbb{P}^2$ be the common relative Jacobian of the Type $i$ KSS pair. The Tate--Shafarevich group is the group of torsors of the Jacobian fibration that are locally tr…

Let $p$ be an odd prime. Let $\mathfrak{T}_p$ denote the set of elliptic curves $E_{/\mathbb{Q}}$ such that the $p$-primary part of the Tate–Shafarevich group…

Fix a prime $p$. For elliptic curves $E/\mathbb{Q}$ ordered by height, consider the proportion of curves for which $p\mid\#\Sha(E/\mathbb{Q})$. Delaunay's proportion conjecture. Th…

Let $K=\mathbb{F}_q(t)$, and for an elliptic curve $E/K$ let $H(E)$ denote its exponential differential height, $N(E)$ its numerical conductor, and $\@math@cyr{\cyracc SH}(E)$ its…

BSD-factor growth conjecture.

For $i\in\{1,2\}$, let $f_k(i,X)$ count integers $u\equiv1\pmod 4$ with $|u|\leq X$ satisfying the paper's condition $(**)$, such that $L(E(u),1)\neq0$ and…

For positive integers $m$ and $k$, consider elliptic curves over the rationals and, for a prime $p$, their $p$-twists. Folklore conjecture. For any positive integers $m$ and $k$, t…

Let $E$ be an elliptic curve over $\mathbf{Q}$, let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis, let $P_K\in E(K)$ be the Heegner point, let $C$ be the pr…

Delaunay's moment conjecture. For every positive integer $\ell$, partition $\lambda=1^{m_1}2^{m_2}\cdots \ell^{m_\ell}$, and nonnegative integer $u$, the average over…

Tate–Shafarevich moment conjecture. The average of

Let $A/\mathbb{Q}$ range over all abelian varieties over $\mathbb{Q}$, and suppose that $\sha(A/\mathbb{Q})$ is finite so that its order has a well-defined non-square part. Stein's…

Let $K$ be a number field and let $E$ be an elliptic curve defined over $K$. The group of $2$-torsion points in the Tate–Shafarevich group is $\Sha(E/K)[2]$, viewed as an…