28 problems

Let $H/F$ be a finite abelian CM extension of a totally real number field $F$, let $G={\rm Gal}(H/F)$, let $p>2$ be prime, and let $S,T$ be as in the source. Write…

Let $L/K$ be a Galois CM-extension with Galois group $G$, and let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramified in $L/K$. Let…

Let $L/K$ be an abelian CM-extension with Galois group $G$, and let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramified in $L/K$. Let…

Let ${\Cal E}/F$ be the Galois extension with Galois group $G$, let $S$ be a set of primes of $F$ containing the infinite primes and those ramified in ${\Cal E}/F$, and let…

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 $F$ be a real quadratic field, let $H/F$ be the relevant narrow Hilbert class field, let $G=\operatorname{Gal}(H/F)$, let $\mathfrak P$ be a prime above $p$, and 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 $u_2=\sum_{\sigma\in G}u_2(\sigma)\otimes[\sigma^{-1}]\in F_{\mathfrak p}^*\otimes\mathbb Z[G]$ be the element obtained from the Eisenstein cocycle and the cap-product construc…

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 $H/F$ be a finite abelian CM extension, and let $R=\mathbf{Z}[1/2][G]^-$, where $G=\operatorname{Gal}(H/F)$. Write…

Let $F$ be a totally real field and let $H/F$ be a finite abelian CM extension. Let $\theta_{S,T}(H/F,0)$ denote the appropriately smoothed Stickelberger element associated to…

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–Brumer–Stark conjecture. There exists $u_{\mathfrak{p}}\in U_{\mathfrak{p},T}^-$ such that

Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field, let $G=\operatorname{Gal}(H/F)$, and let $S$ contain the archimedean and ramified…

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 $F/K$ be an abelian CM extension of totally real number fields, let ${\bf f}$ be an $\mathcal O_K$-ideal divisible by the conductor of $F/K$, and let $w_F$ be the number of roo…