94 problems

Let $Y(1)$ denote the modular curve, and let $S\subset Y(1)^n$ be an irreducible curve that is not contained in a proper special subvariety of $Y(1)^n$. For a special subvariety…

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

Parity conjecture. The unique invariant theta characteristic on $X(p)$ is always even.

Let $N$ be a positive integer. Let $\mathscr{C}_N(\mathbf{Q})$ be the rational cuspidal subgroup of $J_0(N)$, and let $\mathscr{C}(N)$ be the rational cuspidal divisor class group…

Fix $q$, $r\geq 1$, and $d\geq 1$. Let $L$ range over extensions of $F$ of degree at most $d$, and let $\phi$ range over rank-$r$ Drinfeld $A$-modules over $L$. Poonen's Drinfeld-m…

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $\mathrm{c}_1(E)$ be the Manin constant obtained from a modular parametrisation by the modular curve $\mathrm{X}_1(N)$ assoc…

Fix an even character $\theta:(\mathbb{Z}/Np\mathbb{Z})^{\times}\rightarrow\overline{\mathbb{Q}}_p^{\times}$ and let $\omega$ be the Teichmüller character. Assume…

Quadratic isogenies conjecture. There exists an integer $C$, independent of $K$, such that if $N>C$, then every point

Let $d_1$ and $d_2$ be discriminants such that one of them is fundamental when $k$ is even. For any CM points $z_1$ and $z_2$ of discriminants $d_1$ and $d_2$, let $H$ be the compo…

Let $N$ be a positive integer. Let $X_0(N)$ be the modular curve of level $N$, let $J_0(N)$ be its Jacobian, and let $C_N(\mathbb{Q})$ be its rational cuspidal subgroup. Let $C(N)$…

Let $p$ be prime, let $J_0(p)$ be the Jacobian of $X_0(p)$, and let $\mathcal{C}(p)$ be its rational cuspidal divisor class group. Ogg's cuspidal torsion conjecture. The cyclic gro…

Let $E/\mathbb{Q}$ be an elliptic curve. Its modular degree $m_E$ is the minimal degree of a surjective modular parametrization $X_0(N)\to E$ defined over $\mathbb{Q}$ and sending…

Eisenstein quotient conjecture. The map $\widetilde{\varpi}_{G'}$ factors through the Eisenstein ideal, and $\varpi_{G'}$ induces an isomorphism

Let $\widetilde{\varpi}:H_1(X_1(N),C_{\infty},\mathbf Z_p)_+\to\mathcal K$ be the cyclotomic map, and let $\mathfrak I_{\infty}$ be the Eisenstein ideal acting on the source. Shari…

Derickx–van Hoeij conjecture. There is a modular unit $f$ defined over $\mathbb{Q}$ on $X_1(p)$ such that

Galbraith's conjecture. If $2\leq g^+_N\leq 5$, then $X^+_0(N)$ contains exceptional rational points if and only if

Uniform non-modularity conjecture. The same conclusion holds for every $p\neq 3$.

Let $N$ be a prime level, let $\chi$ be the character used to define $h_\chi$, and let $B_{m,\chi}$ denote the $m$-th generalised Bernoulli number attached to $\chi$. Write … For e…

Rationality conjecture. The sheaf ${\mathcal F}_0(h)$ gives a rational point on the Jacobian if and only if, for each $k$, there exists a nonzero formal power series $R_k(q)$ such…

Let $p=u^2+64$ be prime, and let $E_0$ be the Neumann–Setzer elliptic curve of conductor $p$. The modular degree of $E_0$ is the degree of its optimal modular parametrization from…

Stable supersingular-component conjecture. There is a semistable covering of $X_0(p^3)$ defined over $R_p$ such that, for each such $A$, one connected component of the supersingula…

CM residue-disk placement conjecture. The $j$-invariants of all such curves $E$ lie in $2(p+1)/i$ residue disks inside the corresponding Atkin–Lehner circle, with $(p+1)/i$ disks f…

CM placement conjecture at 13. If

CM placement conjecture at 7. If

Let $g_0^+(N,2)$ denote the dimension of the space of weight-$2$ newforms on $\Gamma_0(N)$, equivalently the genus of the corresponding new modular curve. Surjectivity conjecture.…