135 problems

Let $F$ be a field containing a root of unity of order $p$, and let $G_F(p)$ be its maximal pro-$p$ Galois group. Let $\mathfrak{B}$ be the class of standard building blocks and le…

Let $K$ be an infinite field, let $X$ be a curve over $K$, let $K^{\operatorname{sep}}$ be a separable closure of $K$, and let $G_K$ be the absolute Galois group of $K$. For…

Let $K=\mathbb{Q}_q$ be unramified over $\mathbb{Q}_p$, let $\overline{k}$ be an algebraic closure of its residue field, and let $P=\operatorname{Gal}(K^s/K^\mathrm{tr})$ be the wi…

Let $K=\mathbb{Q}_q$ be unramified over $\mathbb{Q}_p$, with maximal unramified extension $K^\mathrm{nr}$ and separable algebraic closure $K^s$. Let…

Let $p>2$ be a prime and let $F$ be a field containing a primitive $p$-th root of unity. Write $F(p)$ for the maximal $p$-extension of $F$, let $G=\operatorname{Gal}(F(p)/F)$, and…

Let $\mathcal{F}$ be the family of polynomial systems $F$ such that $Z(F)$ is finite and the Galois group of $F$ over $\mathbb{Q}$ is dihedral or bicyclic. Let…

Let $p\geq 5$ be a prime number, and suppose that $K/{\mathbb Q}_p$ is an unramified finite extension. Let $X/K$ be a curve of genus $g\geq 2$, embedded in its Jacobian via a $K$-r…

Goncharov's conjecture. The Lie algebra $\mathfrak g$ is a prounipotent Lie algebra freely generated by elements $z_3,z_5,z_7,\dots$, where $z_m$ has weight $-2m$.

Let ${\cal C}(G)$ be the refined dihedral Lie coalgebra for a finite commutative group $G$, let ${C}_{\bullet}(G)$ be its graded dual, and let ${\rm Gr}^W{\cal G}^{(l)}$ be the wei…

Let $p$ be a prime, let $E_m(N)$ be the arithmetic coefficient module defined in the source, let $M^*_{(m)}$ be the dual modular complex, and let…

Let $L_m$ be the rank-$m$ lattice used to define the modular complex, let $V_m:=L_m\otimes\mathbb{Q}$, let $M^*_{(m)}$ be the dual modular complex, and let $S^{w-m}V_m$ denote the…

Let $p$ be a prime number, let $\boldsymbol{}[p]$ be the group of $p$-th roots of unity, let ${D}^{\Delta}_{\bullet}(\boldsymbol{}[p])$ be the diagonal, depth-equals-weight part of…

Let ${D}_{\bullet \bullet}:={D}_{\bullet \bullet}(\{e\})$ be the dihedral Lie algebra for the trivial group, let $\xi=\xi_{\{e\}}$ be its embedding, and let…

Let $N$ be a positive integer, let $\boldsymbol{}[N]$ denote the group of $N$-th roots of unity, let ${\rm Gr}{\cal G}_{\bullet \bullet}^{(l)}(\boldsymbol{}[N])$ be the associated…

Deligne's conjecture. The $\mathbb Q_\ell$-form

Kummer-faithfulness conjecture. Any finite extension of $\overline{K}(\sigma)$ is Kummer-faithful for almost all $\sigma\in G_K^e$.

Let $P_n(z)$ be the sorted binomial polynomial. For odd $n=2m+1$, the source identifies the relevant target group as the $m$th hyperoctahedral group. Hyperoctahedral Galois group c…

Demuškin maximal pro-$p$ Galois group conjecture. If $G_F(p)$ is a pro-$p$ Demuškin group of rank at least $3$, then $F$ is arithmetically Demuškin.

Demuškin-to-$p$-adic type conjecture. If $\mathcal{G}_F(p)$ is a Demuškin cyclotomic pro-$p$ pair of rank at least $3$, then $\mathcal{G}_F(p)$ has $p$-adic type.

Elementary Type Conjecture in Lego form. Every cyclotomic pro-$p$ pair of Galois type and of finite rank is in $\operatorname{ET}_p$.

Efrat's arithmetical Elementary Type Conjecture. There is such a decomposition where, for each $1\leq i\leq n$, one of the following holds: $G_i\cong\mathbb{Z}_p$; $G_i$ is the dec…

Gras's conjecture. For $p\gg 0$, one has

Massey vanishing conjecture. $G_K$ satisfies the Massey vanishing property.

Let $M_n(T)$ be the number of integer $n\times n$ matrices $A$ such that the characteristic polynomial $\chi_A$ of $A$ does not have Galois group the full $S_n$. The notation…

Tame approximation conjecture. There exists a Galois extension $L/\mathbf{Q}$ such that