175 problems

Let $k$ be a perfect field of characteristic $p>0$, let $W(k)$ be its ring of Witt vectors, and set $K=W(k)[1/p]$. Let $f:X\to S$ be a smooth proper morphism of $k$-schemes. For a…

Fix an even character $\theta:(\mathbb{Z}/Np\mathbb{Z})^{\times}\rightarrow\overline{\mathbb{Q}}_p^{\times}$ and let $\omega$ be the Teichmüller character. Assume…

Let $\Delta$ be a lattice polytope of degree denominator $D$, and let ${\rm GNP}(\Delta,p)$ and ${\rm HP}(\Delta)$ denote its generic Newton polygon and Hodge polygon, respectively…

Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum with reflex field $E$, let $G$ be its associated reductive group, and let $\mu$ be the Hodge cocharacter. Let $(r_\mu,V)$ be the re…

Let $A$ be a henselian discrete valuation ring of characteristic zero with residue field of characteristic $p$ and function field $K$. Let $X$ be smooth and projective of relative…

Let convergent and overconvergent $F$-isocrystals be coefficient objects in the relevant $p$-adic cohomological categories, with morphisms considered in the convergent category. Ts…

Let $k$ be a field of characteristic $p>0$, and let an overconvergent $(F,\nabla)$-crystal over $k((t))$ be given. A crystal is potentially semistable if it becomes semistable afte…

Katz's Newton–Hodge polygon inequality. The Newton polygon lies above the Hodge polygon:

Fontaine's ramification conjecture. The group $G_K^{(\mu)}$ should act trivially on $\mathrm{H}^i_{\text{\'e}t}(X_{\overline{K}}, \mathbb{Z}/p^n\mathbb{Z})$ when

Fix a complete discretely valued field $K$ of mixed characteristic $(0,p)$, with ring of integers $\mathcal{O}_K$ and perfect residue field $k$, and let…

Schneider's vanishing conjecture. For all $j>1$,

Let $A$ be a discrete valuation ring with finite residue field $k$, let $X$ be a regular scheme projective and flat over $A$ with strict semistable reduction, and put…

Let $C_f(s,T)$ be the $C$-function of the $T$-adic exponential sums associated with a polynomial $f$, and let its absolute $T$-adic Newton polygon mean the $T^a$-adic Newton polygo…

Let $M_{\theta\mathbb{Q}_p}(n)$ be the concrete $p$-adic realization attached to a Hecke character, and let $G_S$ be the Galois group used in the source. Weak Jannsen conjecture. T…

Let $F$ be a number field, let $\mathfrak{p}$ be a finite place, and let $F_{\mathfrak{p}}$ be the completion of $F$ at $\mathfrak{p}$ with algebraic closure…

Let $F$ be a number field with algebraic closure $\overline{F}$, let $X$ be a smooth projective variety over $F$, let $p$ be a prime, and let $S$ contain the places above $\infty$…

Let $k$ be the base field, let $Y$ be a smooth $k$-variety, let $E$ be an overconvergent $F$-isocrystal on $Y$, and let $\operatorname{sp}_{Y+}(E)$ denote its specialization as an…

Vanishing conjecture. $H_k^0 = 0$ for all positive integers $k$. Consequently, $M_k(T)$ is a polynomial for all positive integers $k$.

Let $k$ be a field, let $Y$ be a separated smooth $k$-scheme, let $E$ be an overconvergent $F$-isocrystal on $Y$, and let $\mathrm{sp}_{Y+}(E)$ denote its specialization. An arithm…

Let $\mathcal{V}$ be a complete discrete valuation ring of mixed characteristics $(0,p)$ with perfect residue field $k$. Let $\mathfrak{X}$ be a separated smooth formal…

Let $Y$ be a smooth variety over $k$ (that is, a separated smooth $k$-scheme of finite type), and let $E$ be an overconvergent $F$-isocrystal on $Y$. A morphism…

The Frobenius matrix $F(\Gamma)$ is the matrix arising from the Frobenius action in the recursive computation of the zeta function, with parameter $\Gamma$. At infinity, a function…

Let $X=E_1\times E_2\times E_3$ be a product of three Tate elliptic curves with parameters $q_i$ for $i=1,2,3$. Let $r$ be the rank of the space of integer triples $(n_1,n_2,n_3)$…

Let $X$ be a variety with totally degenerate reduction, and let $N_{1,L}$ and $N'_{1,L}$ be the enriched monodromy operators defined in Section 4.3. Write…

Let $R$ be the ring of integers in an unramified extension $K$ of $\mathbb Q_p$, let $\pi_R\colon X_R\to C_R$ be an elliptic surface over $R$ satisfying (Rat), and define … where…