10 problems

Prime-divisor conjecture. If $g=2$, then

Torsion-order conjecture. The possible orders $|\operatorname{Tors}(A)(\mathbb Q)|$ are precisely the integers listed below:

Let $p$ be a prime, and let $J_1(p)$ be the Jacobian of the modular curve $X_1(p)$. A rational cusp difference is a divisor class represented by the difference of two rational cusp…

Let $p\geq 5$ be a prime, and let $X_1(p)$ be the modular curve with Jacobian $J_1(p)$. The rational cuspidal subgroup is the subgroup generated by divisor classes given by differe…

Let $\mathbb{F}_q$ be a finite field, let $\mathcal{A}=\mathbb{F}_q[T]$, and let $K=\mathbb{F}_q(T)$. For a monic $\mathfrak{n}\in\mathcal{A}$, let $X_0(\mathfrak{n})$ be the Drinf…

Generalised Ogg's conjecture. For every positive integer $N$,

Schweizer's Drinfeld-module torsion conjecture.

Let $K$ be a number field and let $E$ be an elliptic curve over $K$. Uniform boundedness conjecture. The order of the rational torsion subgroup $E(K)_\mathrm{tor}$ is bounded by a…

Ogg's torsion conjecture. If $E$ is an elliptic curve over $\mathbb{Q}$, then its rational torsion subgroup has one of these forms; equivalently, $X_1(N)$ has no $\mathbb{Q}$-ratio…

Let $A$ be the ring of functions regular away from a fixed place at infinity, and let $\mathfrak n\lhd A$ be a non-zero square-free ideal. Let $X_0(\mathfrak n)$ be the smooth proj…