18 problems

Let $S$ be the arithmetic scheme in the source, let $j$ be the relevant open immersion, let $e$ be the indicated cup-product class, and let $A(1)$ be the real Tate-twist object. Co…

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 $G$ be a reductive group, let $K^p\subset G(\mathbb{A}_f^p)$ be a tame level, let $\operatorname{Iw}\subset G(\mathbb{Q}_p)$ be an Iwahori subgroup, and set…

Let $E$ be an Anderson $t$-module, let $P$ be a prime of $A$, and define … and … Also write $\mathscr{U}(E;\mathscr{O}_L)=\exp_E(U(E;\mathscr{O}_L))$. Leopoldt conjecture. The…

Let $K$ be a CM field, that is, a totally imaginary quadratic extension of a totally real subfield $K^+$. For a prime number $\ell$, let … be a partition of the places of $K$ above…

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 $\tilde\pi$ be a non-critical cuspidal representation, let $x_{\tilde\pi}\in\cE_{\cU,h}^\bullet$ be the associated point, and let $\cW$ be the weight space. Write $\ell$ for th…

Let $R_h$ be the universal minimal ordinary deformation ring and let $\Lambda$ be the Hida–Iwasawa algebra. Deformation-theoretic non-abelian Leopoldt conjecture. … This is identif…

Let $\Lambda$ be the Hida–Iwasawa algebra, let $R_h$ be the universal minimal ordinary deformation ring, let $\mathcal{R}_h$ be the associated simplicial deformation ring, and for…

Let $F$ be a number field, let $L$ be a $\mathbb{Z}_p^d$-extension of $F$, and let $G_S(L)$ be the Galois group of the maximal extension of $L$ unramified outside the primes above…

Let $F$ be a number field, let $p$ be an odd prime, let $S$ be the set of $p$-adic and infinite primes of $F$, and let $G_S(F)$ be the Galois group of the maximal algebraic extensi…

Let $\Lambda_E$ be the regular local ordinary weight algebra of dimension $r=\operatorname{rk}\mathrm T$, let $F_{\mathfrak m}^{\bullet}$ be the complex interpolating the localized…

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 number field satisfying the Leopoldt conjecture for every prime $p$, and let ${\mathcal T}_p$ be the torsion subgroup associated with the maximal Abelian $p$-ramified…

Let $F$ be a number field, $p$ a prime, $n$ a positive integer, $U\subset\operatorname{GL}_n({\mathbb A}_F^\infty)$ a good subgroup, and…

Let $K$ be a totally real field and let $p$ be a prime. Let $\Delta$ be the Galois group of the maximal abelian pro-$p$ extension of $K$ unramified outside $p$. Leopoldt's abelian…

The $\ell$-adic regulator conjecture. Define

Let $K_\infty/K$ be a $p$-adic Lie extension with Galois group $G$, let $V$ be a Galois representation, and let $T$ be a $\mathbb{Z}_p$-lattice in $V$. The equivalent conditions of…