57 problems

Tate–Shafarevich conjecture. The group $\Sha(E_{16n^2}/\mathbb{Q})$ is finite. The source notes that this finiteness would imply that…

Fix a prime $p$, and let $\mathcal{S}_p$ be the set of elliptic curves $E/\mathbb{Q}$ such that $\Sha(E/\mathbb{Q})[p^\infty]\neq 0$. Let $\bar{\mathfrak{d}}$ denote upper density…

Let $K=\mathbb{F}_q(t)$ and let $E/K$ be a non-isotrivial elliptic curve. Its Tate–Shafarevich group is … where the product runs over all places $v$ of $K$. Finiteness conjecture.…

Fix a prime $p$ and an integer $t\geq0$. Consider random elliptic curves of rank $t$, and let $\Sha(E)[p^\infty]$ denote the $p^\infty$-torsion subgroup of the Tate–Shafarevich gro…

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, and let $\FSha(E/\mathbb{Q}_\infty)$ be the fine…

In the Section 5 setting, let $M$ be the integer used in the divisibility argument, and let $\alpha_{p,j_*}$ be the multiplicities of bad components. The source singles out…

Conjecture 1.7. There exists a coefficient $x_{p,j}\in\mathbb{Q}$ such that

Let $k=\mathbb Q(\sqrt{\Delta})$ be the quadratic field associated with the Pell conic … and let $\Sha_2(\mathcal C)$ denote the previously defined group. Tate–Shafarevich conjectu…

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

Let $E$ be an elliptic curve over a global field $K$, and write $\Sha(E/K)$ for its Tate–Shafarevich group. Cassels's conjecture. There should exist a canonical bilinear form … tha…

Let $E/\mathbb{Q}$ be an elliptic curve with $j_E=0$ or $1728$, having additive potentially supersingular reduction at a prime $p\geq 5$. Assume that $E$ achieves supersingular red…

Positive-proportion conjecture. The group

Existence conjecture. For every prime number $p>0$ and each integer $g>0$, there exists an absolutely simple abelian variety $A/\mathbb{Q}$ of dimension $g$ such that

Rank-zero joint distribution conjecture. As $d$ ranges over $\mathcal{E}$, the joint distribution of $\log L(\frac12,E_d)$ and $\log(|\Sha(E_d)|/\sqrt{|d|})$ is approximately bivar…

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.

Jetchev–Stein conjecture. Every class $c\in\Sh(E/\mathbf{Q})$ is visible in a quotient of the Jacobian of a modular curve $X_0(M)$ for some $M$; such a class is called modular.

Tate–Shafarevich finiteness conjecture. The group $\Sha(A/{\mathbf{Q}})$ should be finite. This is not known in general; in particular, it is not known for a single elliptic curve…

Let $F/\mathbb{Q}$ be a totally real, tamely ramified, abelian extension with conductor $\mathfrak{f}$, let $G=\operatorname{Gal}(F/\mathbb{Q})$, and let $A$ be the Jacobian variet…

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…

Birch and Swinnerton-Dyer finiteness conjecture. The group $\Sha(E/\mathbb{Q})$ is finite, and therefore $\Sha(E/\mathbb{Q})[3^\infty]$ is finite.

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…

Tate–Shafarevich finiteness conjecture. In general, $\Sha(C_\lambda/\mathbb{Q})$ is finite.

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $s\in\mathbb{Q}^{\times}\setminus\mathbb{Q}^{\times 2}$. Denote by $E_s$ the corresponding twisted elliptic curve and by…

Let a torus fibered Calabi–Yau manifold define an F-theory compactification, and denote the Tate–Shafarevich group of its fibration by the corresponding group of genus-one fibratio…