9 problems

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$, and let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramifie…

Let $K/F$ be a finite abelian extension with Galois group $G$, and let $S$ and $T$ be sets of places as in the construction above. Let $[... ELLIPSIZATION...]$ denote the isomorphi…

Let $L/K$ be the extension and $G={\rm Gal}(L/K)$, let $C=C_{L,S,T}$ be the perfect complex over $\mathbb{Z}[G]$, and for a prime $p$ let $C_p$ be its $p$-adic realization. For an…

Let $H/F$ be a finite abelian CM-extension, let $p$ be an odd prime, and put $r=\#S'_f-1$. Let $\operatorname{Ker}(\theta_V)_H$ be the theta-kernel module, let…

Let $H/F$ be a finite abelian CM-extension, let $G=\operatorname{Gal}(H/F)$, and let $\Sigma,\Sigma'$ be finite sets of places satisfying the hypotheses used to define the perfect…

Let $H/F$ be a finite abelian CM-extension, and write $G=\operatorname{Gal}(H/F)$. Let $\Sigma$ and $\Sigma'$ be finite sets of places of $F$ satisfying (H1), (H2), (H3), and (H4).…

Let $K'$ be a finite extension of $K$, let $\mathcal{A}$ be the chosen $\mathbb{Z}_p$-order in $A$, and define … Here $\Xi(\mathcal{A},T,K'/K)$ is a graded invertible…

eTNC for $h^0(\operatorname{Spec}L)$ over $\mathbb{Z}_p[G]$.

Let $A$ be an abelian variety over a number field $k$, let $F/k$ be a finite Galois extension with group $G$, and let $h^1(A_{/F})(1)$ be the corresponding Tate-twisted motive with…