7 problems

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 $L/K$ 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 $K'$ be a finite extension of $K$, let $\varepsilon$ be an idempotent of $A[\mathcal{G}_{K'}]$, and let $\mathcal{R}$ be a $\mathbb{Z}_p$-order in…

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