39 problems

Let $G\leq S_n$ be a transitive permutation group. Define the index function by … Let $N(G,X)$ count degree $n$ number fields with Galois group $G$ and absolute discriminant bounde…

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.…

Weak Malle conjecture for pushforward discriminants. For every $\epsilon>0$,

Malle-type conjecture for algebraic tori. For every $n\geq 1$ and every finite subgroup $1\neq H\leq\operatorname{GL}_n(\mathbb{Z})$, there are positive integers $a(H),b(H)$ and a…

Refined Malle conjecture for ${\rm Heis}_4$. The group ${\rm Heis}_4$ fails to satisfy independence of local events, and

Loughran–Santens conjecture. For any balanced counting function $C$, the leading constant in Malle's conjecture is equal to a finite sum of Euler products indexed by certain Brauer…

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 $k$ 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…

Malle's prediction. If $\operatorname{Aut}_{\overline{K}}(C)\cong D_4$, then

Let $G$ be a transitive permutation group, $N\trianglelefteq G$ a normal subgroup, and $Q$ a global field. For a counting invariant $\operatorname{inv}$, let $b(\pi,\phi)$ be the o…

Let $G\subset S_n$ be a transitive permutation group and let $k$ be a number field. Set … Consider the finitely many continuous surjective pairs $(\pi,\phi)$ satisfying the cycloto…

Let $G\subset S_n$ be a finite group and let $Q$ be a global field with $(|G|,\operatorname{char}(Q))=1$. Set … For a cyclotomic subfield $F\subset Q(\mu_d)$, a surjection…

Let $Q$ be a global function field and let $G\subset S_n$ be a transitive permutation group with $(|G|,\operatorname{char}(Q))=1$. Let $\operatorname{inv}$ be a counting invariant.…

Let $k$ be a number field and let $G\subset S_n$ be a transitive permutation group. Let $b_M(G,k)$ be Malle's logarithmic constant, $b(G,k)$ the actual logarithmic constant, and…

Let $G\subset S_n$ be a transitive permutation group and let $Q$ be a global field with $(|G|,\operatorname{char}(Q))=1$. Let $b(G,Q)$ denote the relevant logarithmic exponent and…

Let $G$ be a finite permutation group, let $T\normal G$, and let $q:G\to G/T$ be the quotient map. A group $G$ is concentrated in $T$ when the elements of minimal index in the rele…

Let $k$ be a number field, $G$ a transitive permutation group of degree $n$, and $T\normal G$ a proper normal subgroup. Let $q:G\to G/T$ be the canonical quotient map, let $G_k$ be…

Malle–Bhargava heuristics. One expects

Let $K$ be a number field, let $G$ be a finite group, and let $\mathcal X=BG_K$. Let $c$ be the raising function on the connected components of the relevant inertia stack, and defi…

Let $K$ be a number field, let $G$ be a constant group scheme over $K$, and let $\mathcal{X}=BG_{\mathcal O_K}$ be equipped with a vector bundle $\mathcal V$. For an open substack…

Let $K$ be a number field, let $G\subset S_n$ act transitively on $\{1,\dots,n\}$, and let $M(K,G;B)$ count degree-$n$ extensions $L/K$ with Galois closure of group $G$ and discrim…

Let $K$) be a number field with algebraic closure $\overline K$. Let $G$ be a subgroup of $S_n$ acting transitively on $\{1,\dots,n\}$. For $B>0$, let $M(K,G;B)$ count degree-$n$ e…

Twisted Malle conjecture. There should be a constant $c(K,T(\pi))>0$ such that

Let $G$ be a non-trivial finite group and let $k$ be a number field. Write $\mathfrak{f}(K/k)$ for the product of the primes of $k$ that ramify in $K$. For…