21 problems

Let $G\,|\,V$ be a polar representation, let $f:V\to Q$ be the quotient map, and set … An $A_f$ stratification is the stratification condition described in Section 2.2.1. The Af st…

Let $X$ be a variety over $R$, let $Y$ be a closed subscheme of the special fiber $X_0$, and let $\mathcal{F}$ be an étale constructible sheaf of abelian groups on $X\times_R K$. W…

Loeser's generalized monodromy conjecture. For almost all finite places $\mathfrak P$, if $H$ is a polar hyperplane in $\mathcal P(Z_\phi(F_{\mathfrak P},\chi;s))$, with $\phi$ res…

The vanishing conjecture.

Let $K/E$ be the extension introduced in the construction, let ${\cal M}_{\bf s}$ be the multiplicity space occurring in the decomposition of the nearby-cycle complex, and let…

Gabber–Abe conjecture. If $(f,K)$ is Ψ-good, then $(f,R(\Psi_f)_u^tK)$ is Ψ-good and $(f,K)$ is Ψ-factorizable.

Let $X$ be a flat relative surface over $S$ with an isolated rational singularity on the special fiber, and let $r$ be the dimension of the stalk of…

Let $\mathcal O$ be a strictly henselian discrete valuation ring with fraction field $K$ and residue field $k$, and let $X$ be a proper scheme over $\mathcal O$. Let…

Let $k$ be separably closed, let $\Lambda$ be an algebraic extension of $\overline{\mathbb Q}_\ell$, and let $X\to\mathbb A^1_k$ be a morphism. For a semisimple perverse sheaf…

Let $N = \sum a_y y$ for nonnegative rational numbers $a_y$. Consider the nearby-cycles sheaves $S_V \boxtimes \zeta$ on restricted shtukas $\operatorname{Cht}\mathcal{R}_{G,N,I}$…

Let $K$ be a local field with residue characteristic $p$, let $\ell\neq p$ be a prime number, and let $\mathcal X$ be an admissible formal $\mathcal O_K$-scheme with smooth generic…

Let $K/\mathbf{Q}_p$ be a finite extension with residue field of cardinality $q$, let $C/K$ be a completed algebraic closure, and let $X$ be a qcqs rigid space over $K$. Fix a prim…

Let $C\rightsquigarrow C_0$ be a degeneration of a smooth complex projective curve to a nodal curve, with marked points $S\rightsquigarrow S_0$ disjoint from the nodes. Let…

Let $S$ be a strictly henselian trait with algebraically closed residue field, and let $X$, $x$, $\mathcal F$, and $\mathcal F'$ be as in Theorem: $X$ is a strictly local $S$-schem…

Test function conjecture. As elements of $\mathcal Z(G(E_0),\mathcal G(\mathcal O_{E_0}))$,

Let $\mathcal{Z}^{\vee}\in\textbf{DM}_{\text{gm}}(\mathbb{Q},\mathbb{Q})$ be the dual limit mixed motive, and let $\mathbf{Z}^{\bullet}$ be the complex in…

Let $\mathcal G$ be the Bruhat–Tits parahoric group scheme over $\mathbb Z_{p^r}$ with generic fiber $G_{\mathbb Q_{p^r}}$ and $\mathcal G(\mathbb Z_{p^r})=K_{p^r}$. Let…

Let $Sh_{K_p}$ be the Shimura variety, let $\mathcal M_{K_p}/\mathcal O_E$ be a natural integral model, and let $R\Psi_x$ denote the nearby-cycles complex at a point $x$ of its spe…

Let $F\supset\mathbf{Q}$ be an imaginary quadratic extension with ring of integers $\mathcal{O}$, let $D$ be a central division $F$-algebra with a unitary involution $*$, and let…

Let $G$ be a split reductive group, let ${\rm M}^{\rm loc}$ be the corresponding local model over the ring of integers ${\mathcal O}_E$ of the local reflex field, and let…

Finiteness conjecture. In the analytic case, the same sum is finite in a neighbourhood of any compact set.