21 problems

Let $k_\infty/k$ be a $\mathds Z_p$-power extension with no finite place splitting completely, let $K_\infty=Kk_\infty$, and use the source's decomposition and character $\chi$ to…

Mazur–Rubin–Sano conjecture. In the module

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

Let $k$ be a totally real number field of degree $d$, let $K/k$ be an abelian CM extension with complex conjugation $c$, let $S$ contain the archimedean places, the places ramified…

Rubin--Stark conjecture. If $H^1_\Sigma(\mathcal O_{K,S(K)},\mathds Z_p(1))$ is $\mathds Z_p$-torsion free, equivalently if…

Fix a subset $\Sigma\subseteq\Sigma_S(E)$ satisfying the source's condition, set $r=|\Sigma_S(\mathcal{K}_\infty)|$, $r'=|\Sigma|$, and…

Let $\mathcal{K}_\infty/K$ and $S$ be as in the source, put $\Sigma=\Sigma_S(\mathcal{K}_\infty)$ and $r=|\Sigma|$, and let $\Lambda(\mathcal{G}_\infty)$ be the relevant Iwasawa al…

Let $L/K$, $\Pi$, and $\Pi'$ be the fixed data, let $Sigma$ be a subset of $Sigma(L)$ with $Sigma e Pi$, and let $p$ be a prime such that $O^times{L, Pi, Pi',p}$ is torsion-fre…

Higher-rank Iwasawa main conjecture divisibility. The reflexive hull of the image of the Rubin–Stark element should satisfy

Let $K_\infty/k$ be the relevant $mathds{Z}_p$-extension with Galois group containing $\Gamma$, let $\Sigma$ and $T$ be the prescribed sets of places, and let $V'\setminus V=W$ hav…

Let $L_n/K$, $S$, and $\Sigma$ be as in the source, let $r_T$ be the specified rank, and let $\varepsilon_{L_n\mid K,S,\Sigma}$ be the $r_T$-th Rubin–Stark element. Let…

Let $k_{\Gamma}/k$ be a $\mathbb{Z}_p$-extension with Galois group $\Gamma$, let $\Gamma_n:=\operatorname{Gal}(k_n/k)$, and put $L_\Gamma:=k_\Gamma L$. Let…

Iwasawa-theoretic Mazur–Rubin–Sano conjecture. There exists a unique

Local explicit Iwasawa main conjecture. For every height-one prime ideal $\mathfrak{p}$ of $\Lambda$,

Interpolation conjecture for $\varphi$. There exists a $\Lambda$-basis $\lambda_\varphi\in Q(\Lambda)$ of $\mu_\varphi(\det_\Lambda(C_{K_\infty,S,T}))$ such that, whenever…

Determinant-module Iwasawa main conjecture.

The p-component Rubin–Stark conjecture.

Let $\mathcal{S}^{E,p}_{\infty}$ be the cyclic $\Lambda$-module generated by the tower of Rubin–Stark elements after localization and twisting, let…

Let $F_\infty/F$ be the tower in the paper, let $M=F_{m,n}$ range over its finite subextensions, let $T$ be the representation used in the canonical Selmer conditions, and let…

Let $F$ be a CM field, let $T_\chi$ be the representation attached to $\chi$, let $\Lambda_{\mathcal{W}}$ be the completed Iwasawa algebra, and let…

Let $F$ be a CM field with $[F:\mathbb{Q}]=2g$, let $F_\infty$ be its maximal $\mathbb{Z}_p$-power extension, and let $\Gamma=\operatorname{Gal}(F_\infty/F)$. Let…