94 problems

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.…

Let $X_1(N)$ be the usual modular curve over $\mathbb{Q}$, and call a curve $X$ over $\mathbb{Q}$ modular if there is a nonconstant morphism $X_1(N)\to X$ over $\mathbb{Q}$ for som…

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

Let $k$ be any field and let $E$ be an elliptic curve over $k$ with … Write $E(k)_\mathrm{tors}^\mathrm{is}$ for the intrinsic subgroup, and let $H$ be the subgroup associated with…

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…

Zywina's generalized conjecture. Then

Galois Images Parametrization Conjecture. There exist finitely many explicit open subgroups $G_1,\ldots,G_n$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ such that, for every non…

Let $N$ be the level parameter, let $\alpha$ be the invariant, let $\operatorname{image}\alpha$ denote its image, let $\Phi_m^{\alpha}$ be the modular polynomial with level structu…

Let $N$ be the level parameter, let $\alpha$ be the invariant defining the modular polynomials with level structures, and let $J_1$ be the corresponding polynomial. For a polynomia…

Noncongruence conjecture. Every nonabelian finite simple group $G$ is noncongruence.

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

Let $j \in \Q$ be a non-CM rational $j$-invariant. Write $\cG_j$ for the associated adelic Galois-image subgroup, let $(\cG_j)_{\operatorname{agr}}$ be its agreeable closure, and l…

Let $X$ be a bielliptic modular curve of genus $2$ and level $N$, and write $Jac(X)$ for its Jacobian. A mock rational point over $\mathbb{Q}(\sqrt{D})$ is a point of the type cons…

Let $p$ be a prime. Write $X_0(p)^+$ for the quotient of the modular curve $X_0(p)$ by its usual involution. Uniqueness conjecture. Up to isomorphism, the quotient map … is the onl…

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…