21 problems

Let $E_7$ be an elliptic curve with complex multiplication by $\mathbb{Q}(\sqrt{-21})$, and let … be a fiber of the hypergeometric family. The $(2,3,7)$ Hodge-structure conjecture.…

Let $\pi$ be a self-dual automorphic representation, let $\pi_1=\pi_2=\pi_3=\pi$, and let $f$ range over $\boxtimes_{i=1}^3\pi_i^U$. Vanishing conjecture for equal self-dual repres…

Let $F$ be the totally real field, let $X_U$ be the Shimura curve, and let $\pi_1,\pi_2,\pi_3$ be the automorphic representations and $[\Delta_3]_{f_1\otimes f_2\otimes f_3}$ the c…

Finiteness conjecture. There are only finitely many orbifold $\mathbb{Q}$-subcanonical Shimura curves, up to isomorphism, over $\mathbb{C}$ that are not modular curves.

Effective finiteness conjecture. The set of ideals ${\mathfrak N}$ for which $Q_0({\mathfrak N})_{\mathfrak a}$ is good is an effectively computable finite set.

Finiteness conjecture. There are only finitely many open compact subgroups $K\subset B^\times(\mathbb A_{\mathrm f})$, respectively $K\subset PB^\times(\mathbb A_{\mathrm f})$, up…

Finiteness conjecture. There exist only finitely many congruence Fuchsian groups $\Gamma$ up to conjugation such that $\Gamma\backslash \mathbb H^*$ has vanishing modified diagonal c…

Let $F$ be a field, let $B$ be a quaternion algebra over $F$, and let $g>1$. A curve $C/F$ is $B$-modular if there is a Shimura curve $X_K$ associated with a suitable compact open…

Let $D_1,D_2$ be coprime fundamental discriminants, let $\tau_1,\tau_2$ be the corresponding real quadratic points, and let $p\leq 13$ be prime. Let $J_p(\tau_1,\tau_2)$ be the Dar…

Holomorphic mass equidistribution conjecture. For every such $\Psi$,

Let $F$ be a totally real field, let $D$ be a quaternion algebra over $F$ ramified at exactly one infinite place, and let $\mathfrak{p}$ be a finite place at which $D$ is unramifie…

Let $X_0(\mathfrak{p}_3)$ and $X_0(\mathfrak{p}_3^2)$ be the indicated Shimura curves with models over $\mathbb{F}_{2^3}$. Descent conjecture. The canonical branched cover … and th…

Let a tower of Shimura curves be given, and suppose it has good reduction at a prime. Recursive good reduction conjecture. Good reductions of towers of Shimura curves should be rec…

Let $F$ be a totally real number field of degree $n$ over $\mathbb{Q}$, and let $E/F$ be an elliptic curve modular of Shimura level $1$. Let $\delta_E$ be the relevant degree of a…

Let $F$ be a totally real field, let $D$ be an indefinite quaternion algebra over $F$, and let $S^D(\mathbb F)$ be the direct limit of the mod $p$ étale cohomology of the associate…

Let $F$ be a totally real field, let $D$ be a quaternion algebra over $F$, and let $\overline{r}:G_F\to\mathrm{GL}_2(\mathbb F)$ be a two-dimensional continuous totally odd absolut…

Let $\mathcal{O}$ be the quadratic order used to define the Darmon points, let $H$ be its narrow ring class field over the real quadratic field $K$, and let $J^{(\ell)}_\pm$ be the…

Let $\ell$ be a prime, let $K_\ell$ be the unramified quadratic extension of $\mathbb{Q}_\ell$, and let $T_\epsilon/L_\epsilon$ be the rigid analytic quotient associated with the s…

Let $q$ be a prime number such that $q\not|N\Delta\ell$. Fix a maximal non-Eisenstein ideal $\mathfrak m$ of the Hecke algebra ${\bf T}^{\widehat\psi}(N)$ acting on…

Let $q$ be a prime such that $q\nmid N\Delta'\ell^2$. Let ${\bf T}_0^{\widehat\psi}(N)$ act on $H^1({\bf X}_1(N),\mathcal O)^{\widehat\psi}$, and fix a maximal non-Eisenstein ideal…

Let $S$ be a finite set of primes not dividing $M\ell$. Fix $f\in S_2(\Gamma_0(N\Delta'\ell^2S),\psi)$ that is supercuspidal of type $\tau$ at $\ell$ and special at primes…