170 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…

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

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 a non-CM elliptic curve over $\mathbb{Q}$. For a prime $p$, let … be the residual representation. Serre's uniformity question asks whether there is a constant $N$, depen…

Let … be the restriction of the cyclotomic map $\Pi_N^{\circ}$, and let the corresponding Eisenstein ideal act on its source and target. Sharifi's Eisenstein-ideal isomorphism conj…

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…

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 $d_1$ 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 $p$ 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…

For $N\in\mathbb{N}$, let $S_2^{\rm new}(\Gamma_0(N))$ denote the space of weight-two new cusp forms on $\Gamma_0(N)$, and let $\dim S_2^{\rm new}(\Gamma_0(N))$ be its dimension. M…

Let $\Gamma\subseteq\operatorname{SL}_2(\mathbb{Z})$ be a congruence subgroup of prime level, and let $X(\Gamma)$ be the associated modular curve with Jacobian $J(\Gamma)$. Let…

Let $E$ be an elliptic curve defined over a number field. Supersingular-prime infinitude conjecture. The curve $E$ has infinitely many prime ideals of supersingular reduction. This…

T-classification conjecture. The trace invariant should classify all components of

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

Elkies' conjecture. For all integers $N\gg0$, the points in

Ribet's conjecture. If $\mathfrak{m}$ is an Eisenstein maximal ideal, then

Rademacher's conjecture. The set

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

Modular-form decomposition conjecture. The form $f$ can be written as a sum

Let $p$ be an odd prime, let $V \subset \mathcal{M}_1(\Gamma(4p))$ be the span of the forms $f\mid\alpha$, with $f\in\mathcal{M}_1(\Gamma(4))$ and $\alpha$ ranging over integral ma…

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

Let $D$ be a sufficiently large discriminant, and let $X^{D+}$ denote the Atkin–Lehner quotient of the Shimura curve $X^D$. The folk conjecture. For sufficiently large $D$, the rat…

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…

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…