30 problems

Let $K$ be a CM field that is Galois over $\mathbb{Q}$, and let $\Phi$ be a CM type of $K$. The quantities $p_K(\operatorname{id},\sigma)$ are Shimura period symbols, and…

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 $tilde{boldsymbol{boldsymbol{ obreaktilde{boldsymbol{boldsymbol{ obreakvarepsilon}}}}}}{ obreakmathcal{M},S obreakmathcal{M}}$ be the Rubin–Stark element in…

Stark's conjecture. The quotient $R_{K_0}/R_K$ is the determinant of an $(mn-n)\times(mn-n)$ matrix. This matrix should be diagonalized into $m-1$ blocks, each an $n\times n$ matri…

Assume that $k$ is a CM field. Let $E_p(k)$ be the group of $p$-Weil numbers in $k$, let … where ${\mathbb R}(1)_v$ is the purely imaginary subspace at the complex place $v$, and l…

Stark's conjecture. For every fixed bound on the class number, only finitely many CM-fields have class number at most that bound.

Let $K/F$ be a $p$-ordinary CM quadratic extension for an odd prime $p$, and let $h_K^-$ be its relative class number. Fix embeddings…

Iwasawa main conjecture for $\psi$. In the abelian case, for a finite Hecke character $\psi^{\mathrm{gal}}\colon G_F\to\operatorname{Aut}_{\mathcal{E}}V_\psi$ with $F_\psi$ linearl…

Let $F$ be a totally real field of degree $n$ over $\mathbb{Q}$, and let $K$ be a CM extension of $F$. Write $h_F$ and $h_K$ for the class numbers of $F$ and $K$, respectively, and…

Gross--Kuz'min conjecture.

Let $K$) be a CM field of degree $2n$, and let $\{\sigma_i\}_{i=1}^n$ be a CM type of $K$. The associated Shimura period symbols $p_K(\sigma_i,\operatorname{id})$ are elements of…

Let $g\in\mathbb{Z}_{\geq 1}$. For a CM-field $E$ with maximal totally real subfield $F$, write $\chi_{E/F}$ for the quadratic character associated to $E/F$ and…

Let $K$ be a CM field, let $g=\tfrac12[K:\mathds{Q}]$, and fix a finite-prime-to-$p$ character $\chi:G_K\to\overline{\mathds{Q}_p}^{\times}$ satisfying the conditions in the source…

Let $K$ be a CM-field, let $p\ne2$, and let $R\subseteq K$ be a totally real subfield such that $K/R$ is normal. Set $G=\operatorname{Gal}(K/R)$ and assume $K\cap R_\infty=R$. Let…

Let $K$ be a CM-field and let $p\ne2$. With $A'=\varprojlim A_n'$ and $\Gamma=\operatorname{Gal}(K_\infty/K)$ as above, define the minus part…

Let $M_\Sigma$ be the maximal abelian pro-$p$ extension of $Lk(p^\infty)$ that is $\Sigma$-ramified, let $X_\Sigma:=\operatorname{Gal}(M_\Sigma/Lk(p^\infty))$, and let…

Let $p$ be an odd prime, let $E$ be a CM field of degree $2d$, and let $F$ be a finite abelian extension of $E$ of degree prime to $p$ containing the $p$th roots of unity. Let $K$…

Let $E$ be a CM field with Galois closure having group $G^{\mathcal{CM}}$, and let $H^{00}(G^{\mathcal{CM}})$ be the space of class functions under consideration. Let $h$ be the he…

Let $K$ be a CM-field abelian over a totally real field $F$, with conductor $\mathfrak f_{K/F}$, and assume that $\mathfrak p_\iota$ splits completely in $K/F$. Let $\chi'$ be an o…

Gross's $p$-adic conjecture. One has

Let $K$ be a CM Galois extension of $\mathbb Q$, let $K^+$ be its totally real subfield, and let $\tau\in\operatorname{Gal}(K/K^+)$ be the nontrivial element. Let $\rho$ be an irre…

Let $F$ be a totally real number field of degree $n$, let $E/F$ be a CM extension, and for each CM type $\Phi\in\Phi(E)$ let $X_{\Phi}$ be a CM abelian variety of type…

Let $E$ be a CM field, let $K$ be the extension defined from a prime-to-$p$ character, let $\mathfrak{X}$ be the unramified-outside-$p$ Iwasawa module over $K$, and let…

Let $F$ be the CM field, let $F^+$ be its maximal totally real subfield, let $u$ be the Hecke character giving rise to the Hilbert cuspform $u(theta)$, and let…