4 problems

Let $p$ be an odd prime, let $K/k$ be a finite abelian extension of number fields with Galois group $G$, and let $S$ contain $S_p\cup S_\infty\cup S_{\operatorname{ram}}(K/k)$, wit…

Let $\boldsymbol{k}$ be a perfect field, let $R$ be a $\mathbb{Q}$-algebra, and let $\mathrm{DM}^{\mathrm{eff}}_{\acute et}(\boldsymbol{k},R)$ be the effective étale motivic catego…

Let $K\in\mathcal{F}^+$ be unramified outside a finite set of places $\Sigma$, and let $C_K$ be the associated compactly supported arithmetic complex. Generalized Tate-motive main…

Let $a$ be the tangential base point $(0,\partial/\partial t_{|0})$ for $X:=\mathbb{P}^1_{\mathbb{Q}}\setminus\{0,1,\infty\}$. Let $G_{BK}(\mathbb{Q})$ be the relevant Bloch–Kato m…