12 problems

Let $Y$ be an algebraic manifold of dimension $n$, let $X$ be a completion of $Y$, and let $D$ be an effective boundary divisor with support $X-Y$. For integers $i,j\geq 0$, write…

The coherent cohomological dimension conjecture. One conjectures that

The coherent cohomological dimension conjecture. It is conjectured that

Let $(\pi,\pi')$ be a coherent pair linked by the chain of diagrams constructed in the paper to an elementary pair $(\sigma,\sigma')$. Coherent-pair cohomology conjecture. This cha…

Let $(\pi,\pi')$ be a coherent pair as defined in the paper, and let $(\sigma,\sigma')$ be an elementary pair associated with it. Let $[F]$ and $[F']$ denote flat automorphic vecto…

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 $\Pi=\Pi_f\otimes\Pi_\infty$ satisfy the paper's Assumption, with $\Pi_\infty$ a non-discrete-series representation that is not a discrete series. Set … Regard both as…

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…

Integral lifting conjecture. The resulting action of $U_f^\vee\otimes k$ on the special fiber lifts to an integral action of $U_f^\vee$.

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…

Let $\kappa$ be a dominant weight for $M_\mu/Z_s(G)$, let $V_\kappa$ be the corresponding Weyl representation over $\mathcal{O}_{E',\lambda'}$, and let…

Let $K$ be a complete discretely valued field and let $X$ be a smooth proper variety over $K$. For a nonnegative integer $m$, let $\Gamma^m_{\mathbb Q}(X)$ denote the rational Berk…