81 problems

Tate conjecture. The representation $\rho_{\ell}$ is semisimple, and the cycle class map

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…

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

Let $X$ be a smooth proper variety over a field finitely generated over its prime field, let $H_\ell=H^{2i}(X,\mathbb{Q}_\ell(i))$, and let $G_\ell^0$ be the identity component of…

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…

Tate's strong form of the Tate conjecture. The order $\operatorname{ord}_{s=j}\zeta(X;s)$ at the pole $s=j$ equals

Perfectness conjecture. The pairing is conjectured to be perfect.

Extension-class conjecture. For any $c\in(H^{2p}(X_{\overline{k}},{\mathbb Q}_l(p))^D)^{G_k}$, the extension class $e(c)$ belongs to the image of (0.3).

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-…

Let $X$ be a smooth projective variety over $\mathbb Q$, let $k$ be a number field, and let $q$ be a codimension. Define the Tate-cycle space … and its rank by … Using the analytic…

Let $K$, $S$, and ${\mathcal X}_S$ be as in the function-field setting, let $q-2a=1$, let $B^a(X)=CH^a(X)/CH^a_{\mathrm{hom}}(X)$, and let $z^a_S$ be the sum of the local cycle cla…

Let $X$ be as in the $S$-integral Beilinson setting, let $S$ be a finite set of places containing all Archimedean places, let $q-2a=1$, let $B^a(X)=CH^a(X)/CH^a_{\mathrm{hom}}(X)$,…

Nagao–Tate relation. Nagao's conjecture for $f$ predicts that the first residue in the displayed calculation is the rank of $\mathcal E(K_C)$, while Tate's conjecture for…

Silverman's analytic form of Nagao's conjecture.

Let $k$ be a field and let $\bar{X}$ be the base change of a smooth projective $k$-variety $X$ to an algebraic closure. Write $H^*(\bar{X})$ for its total cohomology, viewed as a m…

Let $Y$ be the Barth–Nieto quintic and let … be the fibre product of the elliptic modular surface $S_1(6)$ over the modular curve $X_1(6)$ with itself. A correspondence between $Y$…

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…

Let $L\text{ be a subfield of }\mathbb{C}$ finitely generated over $\mathbb{Q}$, let $X$ be a smooth projective variety defined over $L$, and let $\ell$ be a prime number. Consider…

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 $X$ be a projective smooth scheme over a finite field $k$, let $\ell$ be different from the characteristic of $k$, and write…

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…