3 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 $S$ be a finite set of places of $k$ containing $S_\infty$ and all places ramified in $F/k$. Let $\lambda:Y_r\to X_r$ be a homomorphism with finite kernel and cokernel, let…

The $p$-part of the Tamagawa number conjecture. The element $T\Omega(\mathbb{Q}(m)_{F'/F})_p$ vanishes: