16 problems

For a field $k$, say that it is $C_2$ if, for every form $F\in k[X_1,\ldots,X_n]$ of degree $d$ with $n>d^2$, the equation … has a nontrivial solution in $k^n$. Artin's conjecture.…

Let $K$ be a number field with finite fourth level, and let $s_4(K)$ denote the least number of fourth powers of elements of $K$ whose sum is $-1$ when such a representation exists…

Toric $p$-adic section conjecture. This map is bijective. The statement is supported by the paper's proof of the corresponding result for torsors under abelian varieties over finit…

Let $F$ be the function field of a $p$-adic curve, let $G$ be a connected linear algebraic group over $F$, and let $Z$ be a projective homogeneous space under $G$. Colliot-Thélène–…

Raskind–Spiess conjecture. The group $T(Y)$ is the direct sum of its maximal divisible subgroup and a finite group.

Let $h(z)$ be a degree-$n$ polynomial over a local field with residue-field size $q$, and let $\rho(n,\sigma;q)$ denote the density of polynomials whose associated finite étale alg…

Let $n$ be a positive integer and let $\sigma$ be a splitting type of degree $n$. For a prime $p$, let $\rho(n,\sigma;p)$ be the Haar measure of the $p$-adic polynomials of degree…

Arboreal Sen Conjecture for PCF maps. If $p \mid d$, then the iterated extension

Divisible-plus-finite conjecture. The group $F^2(X)$ is the direct sum of a divisible group and a finite group.

Let $X$ be an abelian interpretable group. By Corollary, there is an increasing chain of textit{dfg} subgroups $(Y_i:i<\alpha)$ with $Y_0=0$, whose successive quotients have dp-ran…

The dfg conjecture. If $G$ is a $\mathbb{Q}_p$-interpretable zero-dimensional, definably amenable group, then $G$ has dfg.

Let $k$ be a $p$-adic field, and let $F$ be the function field of a smooth projective geometrically integral curve defined over $k$. Let $\Omega$ denote the set of discrete (rank 1…

Definable-isomorphism conjecture. Any infinite field interpretable in $K$ is $K$-definably isomorphic to some finite extension of $K$.

Let $p$ and $t$ be fixed, and let $f\in\mathbb{Z}[x_1]$ be an input $t$-nomial, meaning a univariate integer polynomial with at most $t$ monomial terms. Root-counting conjecture. T…

Let $K$ be a field such that its maximal pro-$p$ Galois quotient $G_K(p)$ is a finitely generated pro-$p$ Demushkin group of rank at least $3$. Elementary type conjecture for Demus…

For a prime $p$, let ${\mathcal F}^*_{n,n+1}$ denote the class of $(n+1)$-nomials in $n$ variables, and let ${\text{{\tt FEAS}}}_{\mathbb{Q}_p}$ be the feasibility problem for rati…