11 problems

Let $X$ be a smooth projective complex variety. Write $CH^2(X)_{hom}$ for codimension-$2$ cycles homologous to $0$, $CH^2(X)_{alg}$ for those algebraically equivalent to $0$, and l…

Let $S^*_{g,i}(j,k)$ be the stratum indexed by $(j,k)$, let $S'_{g,i}(l)$ be a component of the Hecke parameter space, and let $\pi'^*_{i;j,k;l}$ be the restricted projection to…

Let $\mathfrak V$ be a subvariety whose generic point has ordinary reduction at $p$, and let $\tilde{\mathfrak y}$ be the Abel–Jacobi image of its reduction. Let…

Let $S^*_{g}(j)$ and $S'_{g}(l)$ be the partitions of the relevant Hecke parameter space, and let $\pi'^*_{j,l}:S^*_{g}(j)\cap S'_g(l)\to G(j,g-1)$ be the restricted projection. As…

Let $V_l$ be a component in the decomposition of a Hecke correspondence, let $\tilde{\mathfrak z}_l$ be the Abel–Jacobi image of its reduction, and let $\tilde y_1$ be the reduced…

Conjecture 1.7. There exists a coefficient $x_{p,j}\in\mathbb{Q}$ such that

Let $\rho^{?}\in\{\rho,\rho^*(1)\}$, let $K^p\subset \mathrm{H}(\mathbf{A}^{p\infty})$ be open compact, and let $K_p^\circ$ be the chosen hyperspecial subgroup. Let…

Abel–Jacobi MRC conjecture. In this situation, the Abel–Jacobi map always agrees with the MRC fibration for a family of sections.

Bloch–Lewis conjecture. Then $CH^2(X)_{\emph{alg}}$ is finite-dimensional, and consequently

Let $f$ be the normalized newform and let $W_f$ be Deligne's associated representation. For any number field $F$, let … be the map obtained from the $\ell$-adic Abel–Jacobi map, wh…

Let $X$ be a complex manifold, and let $E$ and $F$ be holomorphic vector bundles over $X$ that are $C^{\infty}$-isomorphic. Nadel's relative invariants are denoted by…