40 problems

Let $X_1$ and $X_2$ be varieties defined over $\mathbb{Q}$, and let $\rho_1$ and $\rho_2$ be isomorphic two-dimensional Galois representations occurring in their étale cohomology.…

Let $f:\mathcal{X}\to\mathcal{Y}$ be a fibration in varieties over a number field $K$. Let $R$ be the finite set of excluded prime ideals, let $q_{\mathfrak p}$ denote the residue-…

Generalized Tate conjecture. There is a closed algebraic subset $Z\subset X$ of codimension $k$ such that

Let $k(v)$ be the residue field at $v$, let $\kappa$ be a perfect field over $k(v)$, and let $x\in \mathscr{S}_K(\kappa)$. For each prime $\ell$, write $I_{\ell,x}$ for the corresp…

Generalized Beilinson and semisimplicity conjecture. The cycle class map induces an isomorphism, and the eigenvalue $1$ of $\operatorname{Frob}_q$ acting on…

Jannsen's generalized Tate conjecture. This cycle class map is surjective. It extends the smooth-variety formulation of the Tate conjecture to arbitrary varieties using étale homol…

Let $V$ be a variety with strictly semistable reduction $X$ over a local field with residue field $k$, assume that $k$ is finite, and define … where $\varphi$ is geometric Frobeniu…

Let $s$ be the standard log point over a finite field $k$, let $X$ be a projective vertical log smooth fs log scheme over $s$, and let $E$ be a log Tate curve. Define $K(X,m,r)$ as…

Let $s$ be the standard log point over a finite field $k$, let $X$ be a projective vertical log smooth fs log scheme over $s$, and define $K_0^{\log}(X)$ as the quotient of…

Let $s$ be the standard log point over a finite field $k$, let $X$ be a projective vertical log smooth fs log scheme over $s$, let $m\geq 2r$, and let $E$ be a log Tate curve over…

Let $s=\operatorname{Spec}(k)$ be the standard log point, let $X$ be as above, let $\ell$ be invertible in $k$, and let $m\geq 2r$. Define $KH_{i,\lim}(X)=\varinjlim_{X'}KH_i(X')$…

Let $s=\operatorname{Spec}(k)$ be the standard log point, let $X$ be a projective vertical log smooth fs log scheme over $s$, let $\ell$ be invertible in $k$, and define … where…

Let $s=\operatorname{Spec}(k)$ be the standard log point, let $X$ be a projective vertical log smooth fs log scheme over $s$, let $\ell$ be invertible in $k$, and write … The fixed…

Let $K$ be a real quadratic field, let $I$ be an ideal of ${\mathcal O}_K$, and let $H_{I,K}$ be the Baily–Borel compactification of the Hilbert modular surface of level $I$. For a…

Let $X$ be a smooth projective variety over an algebraically closed field $k$, let $\ell\neq\operatorname{char}(k)$ be a prime, and choose a subfield $k_0\subseteq k$ over which…

Let $V$ be a smooth projective variety over $\mathbb{F}_q$, let $r\geq 0$, and let $A^r(V)$ be the quotient of the space of algebraic cycles of codimension $r$ on $V$ by the subspa…

Let $X$ be a smooth projective variety over a finitely generated field $F$, let ${\mathrm{Ch}}^r_{\hom}(X)$ be the quotient of the Chow group of codimension-$r$ cycles by $\ell$-ad…

Let $X$ be a smooth projective geometrically connected variety over a finitely generated field $K$, let $d=\dim X$, and let $A^i(X)$ be the group of codimension-$i$ algebraic cycle…

Let $X$ be a smooth projective variety over $\operatorname{\mathbb{F}}_q$, let $l$ be a prime, and for each positive integer $n$ consider…

Let $k$ be a perfect field, let $l$ be a prime invertible in $k$, and let $\operatorname{\mathcal{A}}$ be a saturated dg-category over $k$. Let…

Let $X$ be a smooth projective variety over the finite field $k=\operatorname{\mathbb{F}}_q$ of characteristic $p$, with $q=p^m$. Let $W(k)$ be the ring of $p$-typical Witt vectors…

Let $X$ be a smooth projective variety over a field $k$, let $l$ be invertible in $k$, and write $\bar{k}$ for an algebraic closure of $k$. Let…

Assume that the field $k$ is finitely generated, and let $M$ be a motive over $k$. Write $\mathop{\mathrm{G}}_{\ell,k}(M)$ for the $\ell$-adic monodromy group of $M$, and…

Let $F$ be a number field and let $\operatorname{Rep}_{{\mathbb Q}_\ell}(G_F)$ be the category of $\ell$-adic representations of $G_F=\operatorname{Gal}(\overline F/F)$. Consider t…

The $\ell$-adic Tate conjecture for divisors. This map should be surjective. The conjecture predicts that divisor classes account for all cohomology classes fixed by the absolute G…