12 problems

Let $\Pi_{17}$ be the double cover of the Teichmüller curve $W_{17}$, and let its associated supersingular polynomial be the polynomial whose zeros describe the supersingular point…

Let $K$ be a totally real number field, let $X$ be the associated Hilbert modular space with cusp singularities, and let a cusp singularity have a signature defect. Let the Shimizu…

Let $\Pi_{17}$ be the double cover of the Teichmüller curve $W_{17}\hookrightarrow\mathbb{H}^2/\mathrm{SL}(\mathcal{O}_{17}\oplus\mathcal{O}^{\vee}_{17})$. For $j\in\{1,2\}$ and ev…

The motivic action conjecture. There is a rational subspace $V_0\subseteq H^{d_0}(X,\omega)_f$ such that

Let $F$ be a totally real number field of degree $e$, let $m$ be a positive integer, and set $g=me$. Let $\overline{\mathcal A}_F$ be the compactification of the Hilbert modular cy…

Let $F$ be a totally real field in which $p$ is unramified, let $S$ denote the integral model of the Hilbert modular variety with good reduction at $p$, and let $\mathcal{L}$ be th…

Let $F$ be a totally real field, let $G=\operatorname{Res}_{F/\mathbb{Q}}\operatorname{GL}_2$, and let $K\subset G(\mathbb{A}_f)$ be a neat compact open subgroup. Put…

Let $F$ be a totally real field, let $G=\operatorname{Res}_{F/\mathbb{Q}}\operatorname{GL}_2$, and let $Sh_K(G)$ be the Hilbert modular variety associated with a neat compact open…

Archimedean rationality conjecture. The action of $U_f^\vee\otimes E\subseteq U_f^\vee\otimes_\iota\mathbb{C}$ on $H^\ast(X_\mathbb{C},\mathcal{E}_{\underline 1,1})_f$ preserves th…

Motivic action conjecture. There is a degree-shifting action of $U_f^\vee$ such that $H^\ast(X,\omega)_f$ is a rank-one module over $\bigwedge^\ast U_f^\vee$, generated by…

Inert Hilbert modular stratum conjecture. If $w\in W$ has length two, then

Diamond's conjecture. The cone $C_X$ is equal to the cone spanned by the weights of Goren's partial Hasse invariants.