13 problems

Beilinson–Deligne conjecture. The order of vanishing of $\operatorname{L}(\operatorname{Ad}^0(\rho_{f,p}),s)$ at $s=1$ equals

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 $B^a(X)=CH^a(X)/CH^a_{\mathrm{hom}}(X)$,…

Let $X$ be a variety over a number field $K$ with at worst semistable reduction at all places. Let $S$ be a finite set of places containing all Archimedean places, let…

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 $S$ be the generically smooth mixed motive and $D(S)$ its Verdier dual, with compactly supported cohomology, motivic cohomology, weak Hodge realization, and determinant lines a…

With $S=\eta_{!*}\eta^*M$, $D(S)$ its Verdier dual, and the determinant lines and $\mathbb Q$-structures described in the paper, the Beilinson determinant-form conjecture predicts…

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 $U$ be a smooth variety over a field $k$. Define … and … The regulator maps are the maps from $CH^q(U,q)$ to $BH^q(U)$ in the Hodge setting and from…

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…