3 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 $G$. Let $S$ be a finite set of places of $K$ containing all ramified and infinite places, let $\mathcal A_S$ be the module def…

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…