20 problems

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…

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…

Let ${\mathcal{K}}$ be a CM field with maximal totally real subfield ${\mathcal{F}}$, let $p$ be an odd prime unramified in ${\mathcal{F}}$ such that every prime above $p$ splits i…

Let $p$ be an odd prime, let $K$ be a CM field, and let $K^+$ be its maximal totally real subfield. Let $S^+$ be a set of primes of $K^+$ above $p$, each of which splits in $K$ as…

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…

Gross--Kuz'min conjecture.

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 $K$) 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…

The $b7$-Leopoldt conjecture. The map $\log_{\Sigma,p}$ is injective. This is a $b7$-component form of Leopoldt's conjecture and is used to control the non-vanishing of the relevan…

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 $M_\Sigma$ 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 $T$ be the relevant $p$-adic representation, let $e_\chi$ denote the $\chi$-idempotent, and let $H^1_f(G_{k_v},T)$ be the finite local cohomology subgroup at $v$. $\Sigma$-Leop…

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 $K$ be a finite abelian extension of a totally real field $F$ containing a CM-subfield, and let $K_{\mathrm{CM}}$ be its maximal CM-subfield. Put $G:=\operatorname{Gal}(K/F)$,…

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…

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 $K$ be a CM field of degree $2g$, and let $S$ be a primitive CM type for $K$. Let $L$ be the normal closure of $K$, and let $r_K:L^{\times}\to K^{\times}$ be the reciprocity mo…