70 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…

van der Waerden's conjecture. The number of such exceptional polynomials satisfies

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…

Let $G$ be an even group and let $H\leq G$ have index $2$. For a quadratic field $K$, let $f(K)$ count the unramified extensions $L/K$ whose Galois group over $\mathbb{Q}$ is isomo…

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…

Let $C(N,p)$ denote the elliptic curves considered above, and let $p$ range over primes with $p\equiv -1\pmod 4$. Positive-rank conjecture. The curve $C(N,p)$ has positive rank for…

Let $K$ and $K'$ be one-dimensional local fields in the category $\operatorname{LF}(1)=\operatorname{LF}_0(1)\coprod\operatorname{LF}_p(1)$. Write $\Gamma_K$ and $\Gamma_{K'}$ for…

Let $F$ be a field with $\lvert F^\times/F^{\times 2}\rvert\le 8$, and consider possible Galois pro-$2$ groups with three generators arising from absolute Galois groups. Elementary…

Conjecture on $P_{\mathbb{Q},\{\infty,p,l\},p}$. One has

Folk conjecture. For each $n\geq 5$, all but finitely many quadratic number fields admit an unramified extension field of degree $n$ whose normal closure has Galois group $A_n$.

Counting-function filtered $H$-moment conjecture. If $E{Gamma,(Gammav){v otin T}}(Q)neq0$ and $H$ is admissible, then there is a counting function $C$ on…

$E”$-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…

The filtered $H$-moment conjecture. If $E'{Gamma,(Gammav){v otin T}}(D,Q)neq0$ for some $D$, then

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:

Junker–Koenigsmann conjecture. Every characteristic-zero field $K$ with finitely generated Galois group is ample.

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

The infinite-rank conjecture.

Let $C$ be a nonsingular geometrically connected hyperbolic curve over a number field $F$, with function field $F(C)$. Let $\overline{F}^{\mathrm{sep}}$ be a separable closure of…

Bary-Soroker–Kozma's conjecture. With high probability as $n\to\infty$, the Galois group of $f_n$ is the full symmetric group; equivalently, $\Delta(f_n)$ is not a square.

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…

Let $P_n$ be the characteristic polynomial of the tridiagonal random matrix model with $V_1\equiv a$, where $|W_j|$ has a non-degenerate distribution supported on $\mathbb Z_{>0}$…

Let $A\subset\mathbb Z$ satisfy $2\leq \#A<\infty$, and let $\mathcal P_n^{\mathrm{SRM},A}$ be the family of characteristic polynomials of symmetric $n\times n$ matrices with eleme…

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…