8 problems

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 the product of ramified primes be used as the counting invariant for extensions. Product-of-ramified-primes conjecture. This ordering is conjectured to be always suitable for o…

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…

Malle's conjecture. The quantity $|E_k(G,X)|$ has linear growth in $X$ if and only if $G$ contains a transposition.

For each integer $n\geq 2$, let $D_K$ denote the absolute discriminant of a degree-$n$ number field $K/\mathbb{Q}$. The discriminant multiplicity conjecture. For every $\epsilon>0$…

Malle's counting conjecture. For an appropriate constant $c_d$, $N_d(X)\sim c_dX$. This asymptotic is known for degrees $d=2,3,4,5$: classically for $d=2$, by Davenport and Heilbro…

Fix $d+1\geq3$ and a signature $(r_1,r_2)$. Let $L(X)^{r_2}$ be the set of $S_{d+1}$-fields $K$ with $X/2<|D_K|<X$, and let $L(X;\mathcal S)^{r_2}$ be the subset satisfying finitel…