6 problems

Let $F$ be a number field, let $X$ be a smooth projective variety over $F$, and let $M=h^{i-1}(X)(n)$. Write $の\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbbm{1},M)$ for the subspace…

Motivic Lerch element conjecture. The image of this element belongs to

Equivariant Beilinson conjecture. The equality

Let $\pi = \pi_\infty \otimes \pi_f$ be a cohomological cuspidal automorphic representation of $\operatorname{PGSp}_6(\mathbf{A})$. Let $M(\pi_f)$ be the Spin Chow motive with coef…

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 $U$ be a smooth variety over a number field or a local field $F$, and let $\HM^i(U,n)$ be its motivic cohomology. Let $\HMf^i(U,n)$ be the subgroup whose $\ell$-adic Abel–Jacob…