20 problems

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

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 $k$ be a number field and $G$ a transitive permutation group of degree $n$. Write $\mathcal{F}_{\disc,k}(G;X)$ for the set of $G$-extensions $L/k$ with $|\disc(L/k)|\leq X$, wh…

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 $k$ 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 $k$ 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 $G\subset S_n$ 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 $G$ 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…

Let $G$ be a transitive subgroup of $S_n$. Let $\mathcal F_n(X;G)$ be the set of degree-$n$ number fields $K$ satisfying … where $\hat K$ is the Galois closure of $K$ over…

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…

Let $k$ be a number field, let $G$ be a finite group, let $\pi\colon G\to S_n$ be a transitive faithful permutation representation with $n\geq2$, and let $N\unlhd G$ be normal. Def…

Let $G\subset S_n$ be transitive, let $T\normal G$, let $(L/K,\iota_B)$ be a $B$-extension, and let $\pi:G_K\rightarrow G$ be a homomorphism with $\pi\equiv \iota_B\mod T$. Write…

Secondary-term product conjecture. Then the asymptotic distribution of $G_1\times G_2$-extensions has a secondary term of order $X^{c_1/n_2}$. The conjecture proposes a general mec…

Let $K$ be a number field and let $G$ act faithfully and linearly on $V=\mathbb{A}_K^d$, with the associated map $\overline{V}\to\overline{X}=\overline{V}/G$ étale in codimension o…

Let $K$ be a number field, let $G$ be a finite group, and let $V$ be a finite-dimensional faithful representation of $G$ over $K$. Let $N_{G,V,K}(B)$ count isomorphism classes of…

Fix $n\geq 1$. Let $N(k)$ be the number of isomorphism classes of marked degree-$n$ extensions of $\mathbb{F}_q(t)$ with squarefree discriminant of degree $k$, totally split above…

Revised Malle conjecture. Fixing $N$ and $k$, one has