102 problems

Let $X_\infty=(X_n)_{n\ge 0}$ be a geometric $\mathbf Z_p$-tower of smooth projective curves over a finite field of characteristic $p$, with base $X_0=\mathbf P^1$ and ramification…

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 $(\mathsf{G},\mathsf{X})$ 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 $\Delta$ 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…

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 $S$ be a $p$-adic scheme, let $X/S_0$ be a proper SNCL scheme with relative SNCD $D$, and let $f\colon X\to S$ be the structural morphism. Let $P$ and $P^D$ be the weight filtr…

Let $X$ be a variety over $\overline{\mathbb{Q}}$ with good reduction $X_p$ at a prime $p$. Let $Z_0$ be the $\mathbb{Q}$-vector space of algebraic classes on $X$, let $Z_p$ be the…

The Tamagawa number conjecture. There exists a unique $\mathbb{Z}_p$-basis

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 $C_f(s,T)$ 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…

Fix $w\in{}^MW$, and let $\nu:T^c(\mathbb{Z}_p)\to\mathbb{C}_p^\times$ be a continuous character. For a character $\lambda$ of $T^+$ or $T^-$, let $v(\lambda)$ denote its slope in…

Let $w\in{}^MW$, $\kappa\in X^\star(T^c)^{M_\mu,+}$, and let $\chi:T(\mathbb{Z}_p)\to {\overline{F}}^\times$ be a finite-order character. For a character $\lambda$ of $T^+$ or…

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 $\infty$…

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…

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 $X$ 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 $\mathbb Q_p$, and let $\pi_R\colon X_R\to C_R$ be an elliptic surface over $R$ satisfying condition (Rat). Let…

Minimal-valuation local semistable reduction conjecture. $\mathcal{E}$ admits local semistable reduction at any minimal valuation $v$ on $k(X)$ centered on $\overline{X}$.

Let $X$ be a smooth irreducible $k$-variety, let $\overline{X}$ be a partial compactification of $X$, and let $\mathcal{E}$ be an $F$-isocrystal on $X$ overconvergent along…

Let $d\leq 3$, and let $X$ and $Y$ be a maximally generic mirror pair of $d$-dimensional smooth projective Calabi–Yau schemes over the Witt ring $W(\mathbb{F}_q)$. A reduction is g…

Let $X_K$ be a smooth scheme of finite type over $K$. Let $H^{n}_{pst}(X_K)$ be the finite-dimensional $\mathcal{H}^{ur}_1$-module equipped with its increasing weight filtration…

Wan's conjecture. There is a Zariski dense subset ${\mathcal{U}}$ in $\mathbb{A}^d$ such that, for every $f(x)\in {\mathcal{U}}$, the limit exists and

Let $\mathcal{L}$ be a Calabi–Yau operator arising as a Picard–Fuchs operator for a Calabi–Yau motive $X_\varphi$. Let $p\geq5$, and let $\varphi$ be such that…