27 problems

Let $K$ be an $S_n$-field for $n\geq 3$ and signature $(r_1,r_2)$. Let $\mathcal S$ be a finite set of local conditions, and let $|\mathcal S|$ denote the product of their local de…

Let $f$ be the post-critically finite quadratic polynomial under consideration, and let $G_n(f)$ and $M_n(f)$ denote, respectively, its level-$n$ Galois group and even Markov group…

$E”$-family $H$-moment conjecture. The filtered $H$-moment conjecture remains true after every occurrence of $E'{\Gamma,(Gammav){v otin T}}(D,Q)$ is replaced by…

Let … be a polynomial with fixed outer coefficients $a_n=1$ and $a_0=b$, and let $E_{n,b}(H)$ denote the number of such polynomials of height at most $H$ whose Galois group is not…

Bohn–Cameron–Müller's conjecture. The group $G_{K,y}(T_M)$ is maximal, i.e. it is isomorphic to the symmetric group on $\operatorname{r}(M)$ letters:

Let $N_d(X)$ denote the number of number fields of degree $d$ having discriminant with absolute value at most $X$. Let $q(n,k)$ denote the number of partitions of $n$ into at most…

The infinite-rank conjecture. If $G_K$ is finitely generated, then $A$ has infinite rank over $K$.

The infinite-rank conjecture.

Let $f(x)\in\mathbb{Q}[[x]]$ be a $D$-finite series and let $n$ be the minimal order of a differential operator that annihilates $f(x)$. Let $K$ be a number field and let…

Chabert–Halberstadt's conjecture.

Let $K$ be an $S_4$-field. The Chabert–Halberstadt conjecture. … Here $\operatorname{Po}(K)$ is the Pólya group of $K$, and $\operatorname{Cl}(K)$ is its ideal class group. The con…

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 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…

Let $K_i$ be number fields for $i=1,2$, let ${\mathcal C}_i$ be nontrivial full classes of finite groups, and let … be an isomorphism. Here $\Sigma({\mathcal C}_i)$ denotes the ass…

Let $\ell$ be a prime, and let $K$ and $L$ be any fields as in Theorem 2 of the source paper. Write $\mathcal{G}_{K}(\ell)$ for the relevant pro-$\ell$ Galois group. Pro-$\ell$ Gal…

Let $W_{2g}=C_2^g\rtimes S_g$ be the wreath product acting on the $2g$ conjugates in $g$ complex-conjugate pairs, and let $G$ be a transitive subgroup of $W_{2g}$ containing comple…

Let $\underline{\lambda}$ and $\underline{\rho}$ be nontrivial Schubert problems, and let $\underline{\lambda}\circ\underline{\rho}$ be their composition. A composition is fibered…

Let $n\geq 5$ be odd, let $0\neq d\in\mathbb Z$ be chosen so that $g(X)=X^n-X-1+d^2$ has distinct roots, and let $C/\mathbb Q$ be the smooth projective hyperelliptic curve defined…

Let $C/K$ be a smooth, projective, hyperelliptic curve of genus $g\geq 1$ with a $K$-rational point $P_0$, and embed $C$ into its Jacobian $J_C$ using $P_0$. Let $L/K$ be a finite…

Quadratic resolvent counting conjecture. There is a constant $\beta_n$ with $0<\beta_n<1/2$ such that

Let $K|k$ and $L|l$ be function fields over algebraically closed fields. Let $\mathcal{G}_K^a$ and $\mathcal{G}_L^a$ denote the mod-$\ell$ abelianized Galois groups, and let…

Let $k$ be a number field, and let $Gal(\bar{k}/k)$ denote its full Galois group. Absolute Galois group hyperbolicity conjecture. For any number field $k$, the profinite group…

Infinitude conjecture. There are infinitely many imaginary quadratic fields $K$ for which the absolute abelian Galois group is isomorphic to $G$.

Let $K$) be a field that is not locally finite, and let … be topologically finitely generated. Let $E/K$ be a non-trivial elliptic curve. Infinite-rank conjecture. The elliptic cur…