45 problems

Let $\mathfrak p\mid p$ be a place of $F$, with ramification index $e$, residue degree $s$, and inertia group $I_{\mathfrak p}$. Let $W_{\mathfrak p}^?(\rho)$ denote the predicted…

Let $K$ be a number field, let $\mathfrak N$ be an ideal of $\mathcal O_K$, and let $\mathfrak f$ be a non-trivial, new, weight two complex eigenform over $K$ of level…

Let $F$ be a totally real number field, let $K$ be a quadratic imaginary extension of $F$, let $F'$ be a finite totally real extension of $F$, and let $f_{F'}$ be a Hilbert modular…

Let $F$ be a totally real number field, let $f$ be a Hilbert newform of parallel even weight over $F$, let $F'$ be a totally real quadratic extension of $F$, and let $f'$ be the ba…

Let $K$ be a totally real field and let $p\geq3$ be prime. A continuous representation … is geometric if it satisfies the geometricity conditions of Fontaine and Mazur. The Fontain…

Let $F$ be a totally real field in which $p$ is unramified, let $G_F$ be its absolute Galois group, and let $\Sigma_v$ denote the embeddings associated with a place $v\mid p$. Let…

The motivic action conjecture. The classes

Let $r,c\geq 1$ be integers, let $F$ be a totally real field, and let $\rho:\Gamma_F\rightarrow\operatorname{GL}_r(\mathbb{C})$ be a complex representation. The graded space…

Let $f$ be a Hilbert newform of parallel weight $2$ over the totally real field $F$, with Hecke field $K_f$ of degree $g$. Let $s=s_i$ be a sign vector corresponding to the real em…

Let $n=n(m)$, let $\Gamma_m\subset\mathrm{GL}^+_2(\mathbb{R})^n$ be a sequence of groups commensurable with a Hilbert modular group, and suppose that whenever $m\ne m'$ and…

Let $g\in S_2(\Gamma_0(\mathfrak n))$ be a Hilbert modular newform over a totally real field $F$, and let $p$ be an odd prime unramified in $F$ and good ordinary for $g$. Let $M/F$…

Nonsplit quantum unique ergodicity conjecture. For every fixed $H\in C_b(\Gamma\backslash\mathbb{H})$,

Let $A$ be an abelian variety of $GL(2,E)$-type over a number field $K$, with conductor $\mathfrak{n}$, and let $\rho_{A,\lambda}$ denote its Galois representation for a prime…

Let $f$ be a parallel weight-one form, let $U_f^\vee$ be the dual space appearing in the cited construction, and for $u\in U_f^\vee$ let $u_i$ denote its $\sigma_i$-component. Let…

Let $f$ be the Hilbert modular form and let $D_p(f)$ be its $p$-adic de Rham cohomology with partial Frobenius operators $\varphi_1,\dots,\varphi_d$ and partial filtrations…

Span conjecture. Equation $({\mathcal{M}}_F^\theta)^\Delta={\mathcal{M}}_\mathbb{Q}^{(4d/d_2)}$ holds for any totally real field $F$ of degree $d$. This predicts that diagonal rest…

Let $\Sigma_{\infty}$ be the set of archimedean places, let $S\subset\Sigma_{\infty}$, and let $r\in\{0,1\}^{S}$. Let $\pi$, $\tau$, the cohomology classes $\mathbf{g}^{(\delta)}$,…

Let $F$ be the totally real field, let $S$ be the set of places dividing the level and $p$, and let $E/F$ be a totally real abelian extension ramified away from $S$ or at $p$. Set…

Let $F$ be the totally real field, let $S$ be the finite set of places occurring in the construction, and let $E/F$ be a totally real abelian extension ramified away from $S$ or at…

Harris period conjecture. These combinations give rational bases of the corresponding coherent cohomology groups; in particular, $\omega_f^{\Sigma_\infty}/\det R_f$ is rational.

Bloch–Kato conjecture. If $L(\pi,1+j)\ne 0$ for some integer $j$ with $t_i\leqslant j\leqslant k_i+t_i-2$ for all $i$, then

Let $I$ index the relevant embeddings, let $\rX_D^{>r}$ be the boundary region of the eigenvariety, let $\rW^{>r}$ be the corresponding weight-space region, let $w$ be the weight m…

Let $L$ be a totally real number field, let $\ell$ be an odd prime, and let … be an irreducible and totally odd Galois representation, where totally odd means that…

Let $\boldsymbol\alpha$ be a set of parameters from Table 1 and let $\boldsymbol\beta=\{1,1,1,1,1\}$. For a hypergeometric motive $H(\boldsymbol\alpha,\boldsymbol\beta\mid z)$, wri…

Let $g$ be a normalised cuspidal Hilbert modular newform of weight $(k_1,k_2,\ldots,k_d,w=2)$ and level $\Gamma_0(\mathfrak{n})$ over a totally real number field $F$, with each…