7 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 $\textup{BT}$ be the stack of Barsotti–Tate groups over $S$, let $\textup{Disp}$ be the stack of displays over $S$, and let $\phi\colon \textup{BT}\rightarrow \textup{Disp}$ be…

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 $\mathfrak{R}: \textbf{DM}_{\text{gm}}(\mathbb{Q},\mathbb{Q})\to D^b(\textbf{MHS}_{\mathbb{Q}})$ be the Hodge realisation functor. A mixed Hodge-Tate structure is a mixed Hodge…

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…

Fix a field $K$ in the class $\mathcal{F}^+$, unramified outside a finite set of places $\Sigma$, and suppose that $\mathcal{G}:=\mathcal{G}_K$ has no element of order $p$. Let…

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…