181 problems

Mináč–Tân's conjecture. For every field $F$, every prime $p$, and every positive integer $n$, $G_F$ satisfies $n$-Massey vanishing with respect to $p$.

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 $K$ be a finitely generated field, let $V$ be a divisorial set of discrete valuations of $K$, and let $G$ be a connected reductive $K$-group. For each $v otin V$, let $Kv$ deno…

Somekawa's conjecture. The generalized Galois symbol $s_n$ is always injective.

Let $p$ be a prime, let $\mathbb{K}$ be a field containing a primitive $p$th root of $1$, and let $G_\mathbb{K}=\operatorname{Gal}(\mathbb{K}_s/\mathbb{K})$ be its absolute Galois…

Let $K$ be a complete discrete valued field with residue field of characteristic $p>0$, let $L/K$ be a finite Galois extension with Galois group $G$, and suppose that $k_L/k_K$ is…

Let $\mathfrak g=\bigoplus_{m=1}^{\infty}\operatorname{gr}^m\mathfrak g$ be the graded ${\mathbb Z}_p$-Lie algebra arising from the weight filtration on the relevant pro-$p$ Galois…

Serre's Conjecture I. Does

Let $k$ be a number field, let $G$ be a transitive permutation group of degree $n$, and let $N\normal G$. Write $q:G\to G/N$ for the quotient map and … for the induced pushforward.…

Let $p$ be a prime, $K$ a number field, $S$ a finite set of primes of $K$, and $\overline{\rho}:G_{K,S}\to \operatorname{GL}_{n}(\mathbb{F})$ a continuous absolutely irreducible re…

Let $p$ be a prime, let $\mathbb{K}$ be a field containing a root of unity of order $p$, and, if $p=2$, suppose that $\mathbb{K}$ also contains $\sqrt{-1}$. Let $G$ denote the rele…

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 $K=k(C)$ be the function field of a smooth geometrically integral affine curve $C$ over a field $k$, and let $V$ be the set of discrete valuations associated with the closed po…

Let $\mathcal{G}=(G,\theta)$ be a 1-smooth pro-$p$ pair, where $G$ is a pro-$p$ group and $\theta\colon G\to\mathrm{GL}_1(\mathbb{Z}_p)$ is a representation. The group $G$ is weakl…

Eisenstein quotient conjecture. The map $\widetilde{\varpi}_{G'}$ factors through the Eisenstein ideal, and $\varpi_{G'}$ induces an isomorphism

Suppose that $k$ is a field and that $ell$ is a prime distinct from the characteristic of $k$. Let $K/\ell$ be the mod $\ell$ algebraic $K$-theory presheaf of spectra on smooth $k$…

Weigel's Koszulity conjecture. The graded algebra $\operatorname{gr}(\mathbb F_pG_K(p))$ is Koszul. This is a companion conjecture to the expected Koszulity of the cohomology algeb…

Mazur's dimension conjecture. The ring $R$ is flat over $W(k)$ and its relative dimension over $W(k)$ is $\delta(\bar{\rho})$. The conjecture predicts that the expected dimension f…

Serre's Conjecture II. Every such torsor is trivial; equivalently, $P$ has a $k$-rational point.

Let $f$ be a newform of even weight $k$, let $0\leq j\leq k-2$, and assume $L(f,j+1)\neq0$. Let $\Omega^{(-1)^{j+1}}$ denote the relevant period, $u_{j+1}$ the associated unit fact…

Let $p$ be a prime number and $n\geq 3$ an integer. Let $F$ be a field, which contains a primitive $p$-th root of unity if $\operatorname{char}(F)\not=p$. Let $G_F$ denote the abso…

Weak Leopoldt conjecture. The module $\Sha^2_{\mathrm{Iw},S}(V)$ is $\Lambda_{\mathbb{Q}_p}$-torsion.

Let $p$ be a prime with $p\geqslant 3$, let $F^{cyc}$ be the cyclotomic extension of $F$, and let $F^S$ be the maximal extension of $F$ unramified outside the specified set $S$. Fo…

Let $M_{\theta\mathbb{Q}_p}(n)$ be the concrete $p$-adic realization attached to a Hecke character, and let $G_S$ be the Galois group used in the source. Weak Jannsen conjecture. T…

Let $K$ be the imaginary quadratic field and let $M_{\theta\mathbb{Q}_p}(n)$ be the $p$-adic realization of the Hecke-character motive defined in the source, with weight $w$. Let…