30 problems

Let $k_\infty$ be the cyclotomic $\mathds{Z}_p$-extension of $k$, let $K$ be a finite extension in the setup, and let $\mathcal Q_K$ and $\Lambda_K$ be the total quotient ring and…

Let $K/\mathbb{Q}$ be Galois of degree $n$ with group $G$, let $\eta\in K^\times$ generate a multiplicative $\mathbb{Z}[G]$-module of $\mathbb{Z}$-rank $n$, and let…

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 $X$ be the arithmetic object considered in the preceding result, and consider Galois cohomology with restricted ramification, using odd-primary coefficients or working over num…

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 $K/k$ be a CM Galois extension with group $G$ and complex conjugation $c$. Let $U^{-1}$ be the group of minus $p$-units of $K$, let $X^{-1}$ be the minus part of the degree-zer…

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…

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 $\Lambda$ be the Hida–Iwasawa algebra and let $\Sigma_h$ denote the relevant Hida support locus. Hida–Urban's cohomological non-abelian Leopoldt conjecture. … The paper explain…

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 $\mathbb{H}_h^\bullet=e\mathrm{H}^{\bullet}(Y_1(p^\infty),\mathcal{O})$ be the Hida-theoretic cohomology module, and let $q_s$ be the maximal cohomological degree. Hida's conce…

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 $k$ be a number field, let $\ell$ be a prime, and let $m$ be an integer with $m\neq 1$. Write $\mathcal O_k[1/\ell]$ for the ring of integers of $k$ with $\ell$ inverted, and l…

Let $K$ be a number field, let $\Sigma$ be a set of primes of $K$, and let $M$ be a finite extension of $K$. Write $\Sigma_M$ for the set of primes of $M$ above those in $\Sigma$.…

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 $F_0$ be the field in the paper, let $(F_0)_{\mathfrak{P}}$ be its completion at $\mathfrak{P}$, and let $O_{\mathfrak{P}}$ be the corresponding valuation ring. Put … and let…

Let $\Delta$ act on $U_K\otimes_{\mathbf Z}\mathcal F$ and on $\mathcal U_p\otimes_{\mathbf Z_p}\mathcal F$, and let $\chi$ be a character of $\Delta$. Write … for the induced char…

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 $K/\mathbb{Q}$ be Galois, let $G=\operatorname{Gal}(K/\mathbb{Q})$, and let $F$ be the multiplicative $\mathbb{Z}[G]$-module generated by $\eta$. Suppose that $F$ is monogenic.…

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…

The $\ell$-adic regulator conjecture. Define

Leopoldt's conjecture. The elements $i(u_1),\ldots,i(u_r)$ are multiplicatively independent over $\mathbb Z_\ell$; equivalently, the vectors $\log_\ell u_1,\ldots,\log_\ell u_r$ ar…