4 problems

Let $F$) be a real quadratic field, let $_1,\ldots,_\nu$ be distinct prime numbers that split completely in $F$, put $n=_1\cdots _\nu$ (with $n=1$ when $\nu=0$), and let $\zeta_n$…

Let $K'$ be the relevant extension, let $H'$ be its Galois group, let $I(H')$ be the augmentation ideal, and let $r_{K'}$ be the rank of the $S(K')$-unit group. Let $theta_{H'}$ be…

Let $L/K$ be the Galois extension and $G=\operatorname{Gal}(L/K)$, with $G^+=G/\langle\tau\rangle$, and let $e_r$ denote the relevant idempotent. Let $L_{p,S}(r)$ be the central el…

Stark's conjecture $\mathrm{St}(K/k)$. The conjecture $\mathrm{St}(K/k,S,T,S')$ holds for all choices of $S$, $T$, and $S'$ satisfying these hypotheses. This is the global form of…