15 problems

Let $N$ be a prime number. Then $X_0(N)$ has exactly two cusps, $0$ and $\infty$, and let $[0-\infty]$ denote the class in $J_0(N)$ of the divisor $0-\infty$. Ogg's conjecture. The…

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

Let $E$ be an elliptic curve over $Q$. By Mazur's theorem, its torsion subgroup $E(Q)_{\rm tors}$ is one of the fifteen possible torsion groups … or … Here the rank of…

Let $N$ be a positive integer. Let $J_0(N)$ be the Jacobian of the modular curve $X_0(N)$, and let $J_0(N)(\mathbf{Q})_{\mathrm{tors}}$ denote its rational torsion subgroup. Let…

Let $p$ be prime. Let $J_1(p)$ be the Jacobian of the modular curve $X_1(p)$, let $J_1(p)(\mathbf{Q})_{\mathrm{tors}}$ denote its rational torsion subgroup, and let…

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…

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 $N$) be square-free, let $f$ be a weight-$2$ newform on $\Gamma_0(N)$ with integer Fourier coefficients, and let $A$ be the associated optimal elliptic curve. Let $r$ be an odd…

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 $\mathbb{F}_q$ 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…

Flynn's conjecture. There exists a constant $\kappa$, independent of $g$, such that for every $m\leq\kappa g$, there exists a hyperelliptic curve of genus $g$ over ${\mathbb Q}$ wi…

Schweizer's Drinfeld-module torsion conjecture.

Let $A$ be an abelian variety over a number field $K$. Fix the dimension of $A$ and the degree $[K:{\mathbb{Q}}]$. Torsion conjecture. There are only finitely many possible groups…

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…