28 problems

Artin's holomorphicity conjecture. For every nontrivial $\chi\in\mathrm{Irr}(G^+)$, the associated Artin $L$-function $L(s;L/\mathbb Q;\chi)$ is entire.

Let $k$ be a number field and let $G$ be a finite group. Write … The Cohen–Lenstra–Martinet heuristics give a distribution on $\mathbb{Z}[G]'$-modules. Cohen–Lenstra–Martinet conje…

Let $E_n$ be the height-$n$ Morava $E$-theory spectrum, let $G_n$ be its Morava stabilizer group, and let $I$ range over a cofinal collection of ideals defining quotients $E_n/I$.…

For a quartic $S_4$-extension of the rationals, let $d$ denote its absolute discriminant, and let the number of such extensions with discriminant $d$ be counted up to isomorphism.…

Let $L/K$ be a finite Galois field extension, and suppose that $K$ contains a primitive fourth root of unity. The trace form is the non-degenerate quadratic form … defined over $K$…

Bicolinearity conjecture. The map $\ell$ is bicolinear with respect to $(1\otimes\Delta_r)$ and $(\bar{\Delta}_r\otimes 1)$, so that it gives an explicit expression for the strong…

Reduced Boston conjecture. 1. If $G$ is a uniform $p$-adic analytic pro-$p$ group of simple type in characteristic zero and is hereditarily just-infinite, then there is no pro-$p$…

Let $G=D_6\simeq C_2\times S_3$ be the dihedral group of order $12$, and let $\mathcal{F}(X;G)$ denote the Galois extensions $K/\mathbb{Q}$ with $|\operatorname{Disc}(K)|\leq X$ an…

Let $K/\mathbb{Q}$ be a totally real extension whose Galois closure has tetrahedral type, meaning Galois group $A_4$, or octahedral type, meaning Galois group $S_4$. A prime $\ell$…

Let $k$ be a number field, let $G\subseteq S_n$ be a finite group, and let $BG(k)$ denote the groupoid of continuous homomorphisms from the absolute Galois group of $k$ to $G$, mod…

Let $G$ be a finite group, let $K$ be a number field, and let $\operatorname{inv}_1,\ldots,\operatorname{inv}_n$ denote the multi-invariants used to order $G$-extensions $M/K$, wit…

Let $E$ be an elliptic curve over $K=\mathbb{Q}$, and let $p$ be a prime. A $\mathbb{Z}/p\mathbb{Z}$-extension $L/K$ is a finite Galois extension with Galois group isomorphic to…

Let $K$ be a number field, let $G$ be a finite group, let $N\normal G$ be a proper normal subgroup, and let $F/K$ be a fixed $G/N$-extension. Consider $G$-extensions $L/K$ whose fi…

Localized function-field Boston–Markin conjectures. There exists a positive real number $\delta_q^{G,C}$ such that, as $n_1,\dots,n_{d_\lhd(G)}\to\infty$,

Möbius cancellation conjecture. As $X\to\infty$,

Function-field Boston–Markin conjecture. There exists a regular $G$-extension $K/\mathbb{F}_q(T)$, split completely at infinity, such that the number of ramified primes of…

Boston–Markin conjecture. There exists a totally real tamely ramified $G$-extension $K/\mathbb{Q}$ such that the number of finite primes of $\mathbb{Q}$ ramified in $K$ is…

Let $G$ be a finite group, let $K$ be a number field, and let $\operatorname{inv}_1,\dots,\operatorname{inv}_m$ be the invariants of $G$-extensions introduced in the paper. Positiv…

Let $K$ be a number field and let $\Gamma$ be a uniform pro-$p$ group of dimension $d>2$, hence infinite. A Galois extension is finitely and tamely ramified if it is ramified at on…

Let $L/K$ be a given extension of number fields, let $C$ be a conjugacy class in the relevant Galois group, and assume the generalized Riemann hypothesis for the associated Artin…

Local Pre-Malle conjecture. There is a finite set of primes $S_0$ depending on $G$ such that, for every prime $p\notin S_0$ and every integer $k\not\equiv 0\pmod p$, some $G$-exten…

Local Pre-Malle conjecture. There is a finite set of primes $S_0$ depending on $G$ such that, for every prime $p\notin S_0$ and every integer $k\not\equiv 0\pmod p$, some $G$-exten…

Positive-density conjecture. The set $S(G)$ has positive density inside $\mathbb{N}_G$. This conjecture isolates the expected absence of widespread local obstructions to reduced di…

Let $G$ be a reductive group over $F$, let $S$ be a minisotropic torus, and let $E/F$ be a totally ramified Galois extension over which $S$ splits. Let $J\subseteq G(\breve E)$ be…

Let ${\rm K}$ be a number field, and let $\Gamma$ be a uniform pro-$p$ group of dimension $d>2$, so that $\Gamma$ is infinite. A Galois extension ${\rm L}/{\rm K}$ is finitely and…