44 problems

Let $F$ be a number field, let $r_2$ denote the number of pairs of complex embeddings of $F$, and let $F_\infty$ be the compositum of all $\mathbb{Z}_p$-extensions of $F$. Write ……

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 $p$-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…

Let $K$ be a $p$-adic field with residue cardinality $q$ such that $4\sqrt{q}\in\overline{\mathbb{Q}}_\ell4$, let $G$ be a quasi-split reductive group over $K$, and let…

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 $p$ be a prime and let $G$ be a finite group that is potentially $p$-realizable, meaning that it satisfies the ramification-theoretic conditions introduced in the paper. Let…

The p-adic Lévy–Khintchine conjecture. $p$-adic fields have the Lévy–Khintchine property.

Let $i$ and $d$ be integers. A field $K$ is $C_i(d)$ if every homogeneous polynomial of degree $d$ with coefficients in $K$ in $d^i+1$ variables has a nontrivial zero in $K$; it is…

Let $K$ be a local $p$-adically closed field, meaning a finite extension of $\mathbf{Q}_p$, and let $X$ be a smooth complete curve of genus greater than $1$ over $K$. Let…

Let $U$ be a smooth quasi-projective surface over a finite extension $k$ of the $p$-adic field $\mathbb{Q}_p$. Suslin's singular homology $H_0^{\operatorname{sus}}(U)$ has a degree…

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…

Let $F$ be a $p$-adic field, and let $A$ be an algebraic group over $F$. For every finite field extension $K$ of the completion of the maximal unramified extension…

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…

Gazaki–Hiranouchi conjecture. If $v$ is an unramified place of good reduction, then

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…

Generalized density functional-equation conjecture. The density satisfies

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.

Exact value conjecture. If $3\mid d$, then

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.

Artin's additive quartic conjecture. Artin's conjecture holds for all additive quartic forms over ramified quadratic extensions of $\mathbb{Q}_2$. In particular,

Artin's additive form conjecture.