22 problems

Beilinson's Hodge index conjecture. The pairing of the arithmetic theta lift is expected to satisfy

Let $R$ be a regular arithmetic ring, let $B_0$ be a regular integral scheme flat and of finite type over $R$, and let $X_0$ be a smooth projective integral variety of dimension…

Beilinson--Bloch positivity conjectures. For $p\leq \frac{d+1}{2}$, the map

Geometric height pairing and Lear extension conjecture. There is a canonical isomorphism of $\mathbb{Q}$-line bundles over $\overline{S}$:

Let $M$ be a motive over $K$ with coefficient field $\mathcal{F}$, let $\alpha_M$ be its period map, and let $h_M$ be the height pairing on motivic cohomology. Height exact-sequenc…

Let $M$ be a motive over $K$ with coefficient field $\mathcal{F}$, and let $M^*(1)$ denote its dual Tate twist. Let $H_f^1(K,M)$ and $H_f^1(K,M^*(1))$ be the motivic cohomology gro…

Beilinson extension conjecture. For every $\alpha^{p}\in CH^{p}_{\operatorname{hom}}(X)_{\mathbb{Q}}$, there exists an extension $z^{p}_{B}$ on $\mathcal{X}$ such that the image of…

Beilinson extension conjecture. For each $a_{1}\in CH^{p}_{\operatorname{hom}}(X)_{\mathbb{Q}}$, there exists a Beilinson extension…

Gillet–Soulé non-degeneracy conjecture. The Gillet–Soulé height pairing is non-degenerate.

Let $f$ be the family considered above, and let $Z_1$ and $Z_2$ be cycles supported on a single common fiber of $f$. Denote their global Beilinson–Bloch height pairing by…

Let $X_K$ be the Shimura variety considered in the paper, let $\mathrm{Ch}^r(X_K)_{\mathbf{C}}^{\langle p\rangle}$ be the complexified space of codimension-$r$ cycles that are homo…

Height non-degeneracy conjecture. The pairing $\langle\cdot,\cdot\rangle_{\operatorname{GS},\infty}$ is non-degenerate. The conjecture is motivated by arithmetic analogues of the s…

Assume that $\mathcal S/B$ is a smooth family of K3 surfaces, that $\omega_{\mathcal S}=h(S)F$ with $h(S)\in\mathbb R$ the canonical height, and that $\mathcal C$ is the curve in t…

Let $X$ be the fourfold considered in the paper, let $J^2(X)$ be its intermediate Jacobian, and let ${\mathsf L}:J^2(X)\to J^3(X)$ be induced by intersection with an ample cycle. D…

Let $X_K$ be the generic fiber of the arithmetic or geometric family, let ${\operatorname{Ch}}^i(X_K)^0$ denote the subgroup of homologically trivial codimension-$i$ cycles, and le…

Let $B$ be the base and $X$ the smooth proper variety over the relevant function field, with $d=\operatorname{dim}X$ and $p+q=d+1$. Write $CH^p_{\rm hom}(X)_{\bf Q}$ for the group…

Let $a\geq 1$ be an integer and $0\neq d\in\mathbb Q$ such that $f(X)+d^2$ has distinct roots, where $f$ and its roots $\rho,\rho_2,\rho_3$, the splitting field $L_f$, and the elli…

Let $X$ be a smooth projective variety of dimension $d$ over a number field $k$, and let $L_X\in\operatorname{CH}^1(X)$ be the class of a hyperplane section. Assume the hard Lefsch…

Let $X$ be a smooth projective variety of dimension $d$ over a number field $k$. Write $\operatorname{CH}^r_{\hom}(X;\mathbb{Q})$ for homologically trivial codimension-$r$ cycles m…

Let $K$ be a complete discretely valued field, and let the non-archimedean $varphi$-invariant be the invariant introduced earlier in the source. Theorem A gives, for a smooth proje…

Let $X/\mathbb Z$ be projective and equidimensional with smooth non-empty generic fibre $X_\eta/\mathbb Q$, let $d=\dim X$, and use the notation $M=\operatorname{h}^b(X,m)$ and…

Let $k$ be a number field or a function field of positive characteristic, and let $\Delta_\xi$ be the canonical Gross–Schoen cycle associated with the relevant curve or fibration.…