18 problems

Let $F$ be a number field, let $H/F$ be a finite abelian extension with Galois group $G$, and let $S$ and $T$ be disjoint finite sets of places of $F$ such that $S$ contains the ar…

Let $L/H/F$ be a tower in which $F$ is totally real and $H$ and $L$ are finite abelian CM extensions of $F$, with $L$ containing $H$. Write $\mathcal{G}=\operatorname{Gal}(L/F)$,…

Let $F$ be totally real, let $H/F$ be a finite abelian extension in which $\mathfrak{p}$ splits completely, and let $u_2$ denote the Brumer–Stark unit constructed from the second f…

Let $F$ be totally real, let $H/F$ be a finite abelian CM extension with Galois group $G$, and let $S=R\cup\{\mathfrak{p}\}$, where $\mathfrak{p}$ splits completely in $H/F$. Let…

Let $u_3\in F_{\mathfrak p}^*\otimes\mathbb Z[G]$ be the element defined by the cap product … Let $u_{\mathfrak p}$ denote the Brumer–Stark element. Third Brumer–Stark formula. … T…

Let $e$ be the order of $\mathfrak p$ in $G_{\mathfrak f}$, and suppose that $\mathfrak p^e=(\pi)$ with $\pi$ totally positive and $\pi\equiv1\pmod{\mathfrak f}$. Let $\mathcal D$…

Let $u_{\mathfrak p}$ be the Brumer–Stark element attached to the data $(S,T,H,\mathfrak P)$, and let $u_1,u_2,u_3\in F_{\mathfrak p}^*\otimes\mathbb Z[G]$ be the three formulas de…

Brumer–Stark conjecture. There exists $u_T\in\mathcal{U}_{\mathfrak{p}}$ such that $u_T\equiv1\pmod T$ and, for every $\sigma\in G$,

Let $F_p^*$ be the relevant local multiplicative group and let $\widehat{F}_p^*=F_p^*\widehat\otimes\mathbf{Z}_p$. Define $v_p(\mathfrak b)^{\mathrm{an}}\in\widehat{F}_p^*$ from th…

Let $F$ be a totally real field, let $H/F$ be a finite abelian CM extension, and let $\mathfrak p$ split completely in $H$. For a prime ideal $\mathfrak b$ prime to the prescribed…

Let $H/F$ be a finite abelian CM extension, let $R=\mathbf{Z}[1/2][G]^-$, and define the Selmer module … with contragredient $G$-action. Let $\Theta_{S,T}$ be the smoothed Stickelb…

Let $R=\mathbf{Z}[1/2][G]^-$, let $\operatorname{Cl}^T(H)^-$ be the minus part of the $T$-smoothed class group, and for each ramified place $v$ let $A_v=\mathbf{Z}[G/I_v]/(g_v)$. W…

Let $H/F$ be as in the Brumer–Stark setting, let $L/F$ be a finite abelian CM extension containing $H$, and write $\mathfrak{g}=\operatorname{Gal}(L/F)$ and…

Let $F$ be a totally real field, let $\mathfrak{n}$ be a nonzero ideal, let $F(\mathfrak{n})$ be its narrow ray class field, and let $H$ be the maximal CM subfield in which…

Let $F$, $H$, $S$, $T$, and $\mathfrak{p}$ be as above, and let $L/F$ be a finite abelian CM extension containing $H$ and unramified outside…

Tate–Gross conjecture. For every $\sigma\in G$, $u_{S,T,\sigma}\in{\mathscr E}_{\mathfrak p}$; for every $\sigma\in G$ and every ${\mathfrak q}\in T$,…

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. For $\alpha\in L^\times$, define…

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…