39 problems

Let $F$) be a totally real field in which $p$ is inert, let $E/\mathbb{Q}_p$ be a finite extension with ring of integers $\mathcal{O}$, and let…

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 $D$ 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…

Universality conjecture. There exists an automorphism $\sigma\in\operatorname{Gal}(\overline{\mathbb{Q}}/k)$ such that $\mathord{\uOmega}$ and…

Let $M_5$ denote the moduli space of curves of genus $5$, and let an ST-curve mean a curve that is both Shimura and Teichmüller. Nonexistence conjecture for genus 5 ST-curves. Ther…

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…

Let $\boldsymbol{\tau}$ be a point on one of the modular or Shimura curves considered in the source, and let $\mathcal{L}^{lin}_{\boldsymbol{\tau}}$ be its lattice of singular line…

Let $\boldsymbol{\tau}$ be a period matrix on one of the Shimura curves under consideration, and let $\mathcal{L}_{\boldsymbol{\tau}}$ and $\mathcal{L}^{lin}_{\boldsymbol{\tau}}$ d…

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…

Finiteness conjecture. Only finitely many Shimura curves can have vanishing modified diagonal cycles.

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…

Krazer–Rosen's finiteness conjecture. There are only finitely many Shimura curves $X$ defined over ${\mathbf Q}$ such that

Let $C$ be the curve over $F$ considered in the paper, let $K$ be the quadratic field used in its construction, and let $\mathcal{O}$ be the suborder of $\mathcal{O}_K$ of index…

Igusa-level Shimura-curve conjecture. The curves $X$ and $Z$ should be obtained from Shimura curves by an “Igusa level structure” construction.

Fake elliptic curve conjecture. The local system $\mathscr{L}$ should come from a family of fake elliptic curves: there exists a map $f:C\to X^{D}$ and a family…

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…