79 problems

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 $A$ 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 totally real field, let $p$ be unramified in $F$, and let $\rho:\operatorname{Gal}(\overline{F}/F)\rightarrow\operatorname{GL}_2(\overline{\mathbb{F}}_p)$ be a continu…

Let $F$) be a totally real field, let $\mathbf{f}$ be a Hilbert modular form of parallel weight $2$ over $F$, and let $L_p(s,\mathbf{f})$ be its cyclotomic $p$-adic $L$-function. L…

Buzzard–Diamond–Jarvis conjecture. The representation $\rho$ arises from a Hilbert modular form. This is the folklore modularity conjecture underlying the Serre weight conjecture;…

Let $F/E$ be a Galois extension with group $G=\operatorname{Gal}(F/E)$, and let $f$ be a Hilbert newform of $G$-invariant level $\mathfrak{N}$, weight $k$, and character $\chi$. Fo…

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 $X/\mathbb Q$ be the Calabi–Yau threefold considered by Consani and Scholten, with $b_3(X)=4$, and suppose its Galois representation on…

Beilinson–Deligne conjecture. The order of vanishing of $\operatorname{L}(\operatorname{Ad}^0(\rho_{f,p}),s)$ at $s=1$ equals

Let $F=\operatorname{\mathbb{Q}}(\sqrt{5})$, let $\widetilde{X}$ be the proper and smooth $\operatorname{\mathbb{Z}}[\frac{1}{30}]$-scheme described in the paper, and let … Suppose…

Recall the plectic Galois group … where $F$ is the fixed totally real field, $S$ is the specified finite set of places, and $\bar{\mathbb{Q}}^{S-\mathrm{ur}}$ is the maximal subfie…

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…

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…

Plectic Siegel classes conjecture. There exist two distinct nontrivial plectic Siegel classes

Let $\rho:\operatorname{Gal}(\overline{F}/F)\rightarrow\operatorname{GL}_2(\overline{\mathbb{F}}_p)$ be a continuous, totally odd, irreducible representation for a totally real fie…

Hamahata's conjecture. For every elliptic curve $E$ over a totally real number field, there is a correspondence between a Hilbert modular variety and the product of the conjugates…

Let $F$ be a totally real field, let $f$ be a Hilbert newform with Hecke field $K=K_f$, and let $\chi$ be a quadratic character of sign $s\in W_\infty$. Suppose that…

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 $\mathcal{U}$ be a neighbourhood in weight space over which a Hida family $\underline{\Pi}$ is defined, and suppose the associated Eichler--Shimura isomorphism exists after bas…

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 $F$ range over totally real number fields of degree $n$, let $\mathfrak{n}$ range over non-zero integral ideals of $F$, and let $f$, $g$, and $h$ be Hecke eigenforms of level…

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 $M/F$…

Let $F$ be a totally real number field, let $\pi=\bigotimes_v\pi_v\in\mathrm{RA}_{F}\cap\mathfrak{F}_{2}^{*}$, and let $S_{\pi}$ be the finite set of non-archimedean places where…