68 problems

Let $S$ consist of the infinite place and the primes dividing $Np$, let $\mathbb{Z}_S=\mathbb{Z}[1/Np]$, let $\Lambda$ be the cyclotomic Iwasawa algebra, and let $\mathbf{T}$ be th…

Perrin-Riou's conjecture. Equality should hold in this inclusion:

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…

Euler system existence and nonvanishing conjecture. If $V$ is $r$-critical and satisfies the rank $r$ Panchishkin condition for $r\geq 0$, there exists a collection of cohomology c…

Mazur–Rubin–Sano conjecture. In the module

Stark conjecture. The Rubin–Stark element $\eta_{E/K,S(E)}^V$ should belong to the corresponding integral lattice, namely

Let $T_\Pi$ be the Galois lattice attached to $\Pi$, let $\mathcal O$ be its coefficient ring, and let $\mathcal N$ be the set of integers $mp^k$ where $m$ is a square-free product…

Let $\mathcal{F}$ be the module of functions on roots of unity satisfying the relevant domain conditions, and let $\mathcal{F}^{\rm d}\subseteq\mathcal{F}$ be the $\mathcal{R}$-mod…

Let $F$) be a real quadratic field, let $_1,\ldots,_\nu$ be distinct prime numbers that split completely in $F$, put $n=_1\cdots _\nu$ (with $n=1$ when $\nu=0$), and let $\zeta_n$…

Let $U$ be a sufficiently small affinoid in the half-ordinary universal deformation family, let $V_U$ be its associated representation, and let $\mathscr{D}^+_U$ be the chosen Panc…

Let $\mathcal{H}(\Gamma)$ be the relevant Iwasawa algebra, and let $\widetilde{H}^2(\mathbf{Q}_\infty,V,\mathcal{D}^+)$ and…

Let $K_infty/K$ be the anticyclotomic $Z_p$-extension, let $Lambda$ be its Iwasawa algebra, and let … Here $Theta$ is the canonical map from the determinant module of the Selmer co…

Let $\underline{\Pi}$ be a family on $\operatorname{GSp}_4$ with root number $-1$ and no endoscopic specialisations, let $\underline{\sigma}$ be the accompanying $\mathrm{GL}_2$ fa…

Let $\underline{\Pi}$ be a family on $\operatorname{GSp}_4$ whose specialisations have root number $-1$, let $\underline{\sigma}$ be the accompanying $\mathrm{GL}_2$ family, and le…

Plectic Siegel classes conjecture. There exist two distinct nontrivial plectic Siegel classes

Rank-two plectic factorization conjecture. One has

Rank-one factorization conjecture. Up to some $p$-adic meromorphic function, one has

Let $\mathcal{U}$ be a neighbourhood in weight space over which a Hida family $\underline{\Pi}$ is defined, and suppose the associated Eichler--Shimura isomorphism exists after bas…

Let $\mathcal O$ be the coefficient ring, let $I$ be the relevant augmentation ideal, let $c\in\bigcap_{\Lambda_I}^r H^1_\Sigma(\mathcal O_{K,S},\mathbb T)_I$ be the special elemen…

Let $\underline{\pi}$ and $\underline{\sigma}$ be the Coleman families and let $\mathbf{z}_1(\underline{\pi}\times\underline{\sigma},\gamma_S)$ be the associated Euler-system cohom…

Let $T_f$ be a Galois-stable $\mathcal{O}_\pi$-lattice of $V_f$ satisfying Assumption (Im0), and let $r\in\mathbb{Z}$. Assume that $p$ does not divide $N$. For $i=0,\ldots,p-2$, le…

Let $f_\beta=E_{r+2}^{\operatorname{crit}}(\psi,\tau)$ be the critical-slope $p$-stabilisation of a $p$-decent Eisenstein series, and suppose the equivalent conditions in the prece…

Let $k^{\mathrm{s}}$ be the maximal abelian extension of $k$ in which all places in $S_\infty(k)$ split completely, let $\Omega$ be the collection of finite ramified extensions of…

Let $k$ be the number field in the paper, let $S_infty(k)$ denote its set of infinite places, and let $\mathcal{V}$ be a subset of $S_infty(k)$. For a finite set $\mathcal{S}$ of p…

Let $\Lambda_{\mathbf f}$ be the Iwasawa algebra, let $\mathbb V_{{\mathbf f}gh}^{\dag}$ and $\mathbb A_{{\mathbf f}gh}^{\dag}$ be the associated $\Lambda_{\mathbf f}$-adic represe…