289 problems

Sharp upper-bound conjecture. Assume $p=2$. For any Galois representation $\rho$, one has

Let $r\geq 1$. Let $F$ be a CM field, and let $(\pi_1,\psi_1)$ and $(\pi_2,\psi_2)$ be RAECSDC automorphic representations of $\mathrm{GL}_2({\mathbf A}_F)$ and…

Fix an imaginary CM field $F$ in which $p$ is unramified, let $F^+$ be its maximal totally real subfield, and set … Let $({\mathbb Z}^2_+)^S_0$ consist of tuples $a$ satisfying…

Let $p$, the finite field $k$, the deformation data $\mathcal D$, the absolutely irreducible type-$\mathcal D$ representation $\bar\rho:G_{\bold Q}\to\operatorname{GL}_2(k)$, and t…

Let $p$ be an odd prime, let $k$ be a finite field of characteristic $p$, and let deformation data be a pair $\mathcal D=(\Sigma,t)$ consisting of a finite set of primes and one of…

Let $G_{\bold Q}=\operatorname{Gal}(\bar{\bold Q}/\bold Q)$, let $p$ be an odd prime, and let $\bar{\varepsilon}_p:G_{\bold Q}\to\bold F_p^\times$ be the character describing the a…

Let $E$ be an elliptic curve over ${\mathbf Q}$, and let $d\geq 1$ be fixed. A prime $p$ is a local torsion prime for $E$ if there exists an extension $K/{\mathbf Q}_p$ of degree a…

Crystallinity conjecture. If $pi$ is unramified at $p$, then

Let $K$ be a number field and let $X$ and $X'$ be smooth hyperbolic curves defined over $K$. Denote by $\pi_1^{mot}(\bar X)$ the motivic fundamental group over $\bar K$, and let…

Mazur's principle. Suppose that $m$ is not pseudo-Eisenstein and that $N^r=0$ on a representation space for $\overline{\rho}_m$. Then there is a level $U'$ containing $U$, isomorph…

Let $E$ be the quadratic imaginary field in the paper, let $G_E$ be its absolute Galois group, and let $\pi$ be an automorphic representation of $\operatorname{U}(m)$ with regular…

$C_{\mathrm{EP}}(L/K,V)$. The map $\delta_{V,L/K}$ sends $\Delta_{\mathrm{EP}}(L/K,T)$ onto $\mathbf{Z}_p[G]_{V,L/K}$.

Langlands correspondence for $\mathrm{GL}_2/K$. The representation $\rho_{\lambda}$ exists as above, is potentially semistable at every place $v$, and its associated representation…

Let $X_1$ and $X_2$ be varieties defined over $\mathbb{Q}$, and let $\rho_1$ and $\rho_2$ be isomorphic two-dimensional Galois representations occurring in their étale cohomology.…

Let $V$ be a crystalline representation of ${\rm G}_{{\mathbb Q}_p}$, let $T$ be a Galois-invariant lattice, and write $V\bmod p$ for the semisimplification of $T/pT$. Let $\Pi(V)$…

Let $\mathbf{P}\overline{\rho}: \mathrm{Gal}(\overline{\mathbf Q}/\mathbf Q) \rightarrow \mathrm{PGL}_2(\mathbf F_3)$ be an irreducible representation. Assume that…

Let $\mathbb P^1-\{0,1,\infty\}$ be the thrice-punctured projective line over $\mathbb Q$, let $Gal(\overline{\mathbb Q}/\mathbb Q)$ be its absolute Galois group, and let…

Fix an odd prime $p$, a finite extension $E$ of ${\mathbb Q}_p$ with residue field ${\mathbf k}_E$, and a potentially crystalline representation…

Let $\overline{\Omega}$ be the closure of the crystalline locus in the deformation space $X(\overline{\rho})$, and let $\rho \in \overline{\Omega}$ be a classical modular point. Sa…

Let $\phi$ be a Maass form with Laplace eigenvalue $\frac14$. Cassels's conjecture. Must $\phi$ arise from a Galois representation? The source presents this as an open conjectural…

Let $\rho$ be a motivic Galois representation with open image as above, let $m\geq 2$, and let $\mathcal{P}$ be a Cebatorev set, meaning that, up to finitely many primes, it is spe…

Let $\rho:\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to\operatorname{GL}_n\mathbf{Q}_\ell$ be a motivic Galois representation, and let $\delta_m(\rho)$ be the density rat…

Functional-equation conjecture. One has $m_p^R=m_{w-p}^R$ for all $p$, hence $m_{w/2}\equiv\dim V\pmod 2$, and: (1) $L(\iota R,s)$ extends to an entire function except for a single…

Compatible-systems conjecture. The following should hold: (1) every continuous semisimple de Rham representation $R:G_{\mathbb{Q}}\to GL_n(\overline{\mathbb{Q}}_l)$ unramified at a…

Let $S$ be the fixed set of primes and let $S'=s'_1,\dots,s'_n\}$ be a ramified auxiliary set whose existence is guaranteed by Theorem 1. The ring $R_{S\cup S'}$ is a finite flat c…