6 problems

Let $L/K$ be an abelian extension of number fields with Galois group $G$. Let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramified in…

Let $L/K$ be a Galois CM-extension with Galois group $G$, and let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramified in $L/K$. Let…

Let $K/k$ be a finite Galois CM-extension with Galois group $G$. Let $S$ contain all ramified finite places and all infinite places of $k$, let $\goth{A}_S$ be the module generated…

Let $K/k$ be a finite Galois CM-extension of number fields with Galois group $G$. Let $S$ contain all infinite places and all finite places ramified in $K/k$, and let…

Let $K/k$ be an abelian extension of number fields with Galois group $G$, and let $S$ contain the places ramified in $K/k$. For each odd prime $\ell$, let $\mathcal{J}^r(K/k)$ be t…

Let $k$ be totally real, let $K$ be totally real or a CM-field, let $K/k$ be an abelian extension with Galois group $G$, and let $S$ contain the infinite places and the places rami…