3 problems

Let $k$ be a totally real number field, let $p$ be an odd prime splitting completely in $k$, and let $K$ be as in the paper. Write $G=\operatorname{Gal}(K/k)$ and let $\delta(K/k)$…

Let $k$ be a totally real number field, let $p\neq 2$ be a prime, and write ${\mathfrak f}'$ for the prime-to-$p$ part of the conductor ${\mathfrak f}$ of…

Torsion conjecture. For every $K$, $p$, and $S$,