136 problems

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be a smooth projective curve of genus $g$ over $k$, let $B$ be a non-empty set of $r$ points of $X$, and l…

For every prime $p$, every integer $n\ge 1$, and every algebraically closed field $k$ of characteristic $p$, the essential dimension of the cyclic group $C_{p^n}$ is $n$:…

Let $L/K$ be a Galois extension with Galois group … For every irreducible character $\chi\colon G\to\mathbb{C}$, let $L(s,\chi)$ denote the associated Artin $L$-function. Artin's h…

Let $K$ be a number field, let $p$ be a prime, and let ${\mathcal T}_{S_p}$ denote the relevant tame ramification module for the maximal extension of $K$ unramified outside the pla…

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…

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

Let $G$ be a nontrivial finite group. Write $d(G)$ for the minimal number of generators of $G$, $G^{\mathrm{ab}}$ for its abelianization, and $p(G^{\mathrm{ab}})$ for the subgroup…

Let $n\geq 2$, let $Q_n$ be the polynomial studied in the paper, and write … For a polynomial with rational coefficients, let $\operatorname{Gal}(Q_n)$ denote its Galois group and…

Let $G$ be a finite group. A group is Sylow metacyclic if all of its Sylow subgroups are metacyclic. Schacher's conjecture. The group $G$ is $Q$-admissible if and only if $G$ is Sy…

Let $K$ be a finitely generated field over its prime field, let $e$ be a positive integer, and let $\sigma$ range over $\mathrm{Gal}(K^\mathrm{sep}/K)^e$. For almost all such…

Let $p$ be a prime. Gross's conjecture. For every prime $p$, there exists a nonsolvable Galois number field $K$ ramified only at $p$. This conjecture concerns the existence of numb…

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…