28 problems

Let $F$ be a global field, let $G$ be an algebraic group defined over $F$, and let $\tau(G)\in\mathbb{R}$ denote its Tamagawa number, the volume of…

Let $K$ be a global field, and let $K^{\mathrm{cyc}}$ denote its maximal cyclotomic extension. The absolute Galois group of a field $E$ is denoted by $G_E$. Shafarevich's freeness…

Let $K$ be a global field and let $X\subset \mathbb{P}_K^{n-1}$ be a cubic hypersurface defined over $K$. Folklore conjecture. The set $X(K)$ of $K$-rational points on $X$ should b…

Let $K$ be a global field and $F$ a finite normal extension of $K$. Let $C$ be a conjugacy-stable subset of $\operatorname{Gal}(F/K)$, let $d$ be a positive integer (coprime to the…

Extended Schinzel hypothesis H. If

Profinite Bateman–Horn conjecture. The following limits exist and satisfy

Let $K$ be a global field with ring of integers $\mathcal{O}_K$. Let $P_1[x],\dots,P_k[x]\in\mathcal{O}_K[x]$ be nonconstant polynomials, and let $d^*(A)$ denote the upper Banach d…

Let $F$ be a global field of characteristic $p\geq 0$, let $l\neq p$ be a prime, and for each place $v$ let $F_v$ be the completion of $F$ at $v$. Let $X$ be the scheme under consi…

Axiomatizability conjecture. There exists a theory in the language of globally valued fields whose models are exactly the existentially closed globally valued fields of archimedean…

Let $K$ be a global field, let $Y$ be a smooth projective variety over $K$, and let $i\in\mathbb{Z}$. Write $\operatorname{CH}^i(Y)_0$ for the kernel of the cycle class map and let…

Let $Y$ be a smooth projective variety over a global field, and let $i\in\mathbb{Z}$. Write $\operatorname{CH}^i(Y)_0$ for the kernel of the cycle class map on…

Let $K$ be a global field and let $C$ be a smooth projective curve of genus at least two. Mordell–Lang conjecture. The number of $K$-rational solutions of $C$ is finite. This is a…

Global localization conjecture. For $j>0$, there is an isomorphism

Uniformity conjecture. There is a function $f:\mathbb{N}\to\mathbb{N}$ such that, for any $K$, $S$, and $R$ as above,

Let $K$ be a global field and let $v$ be a place of $K$. Let $X$ and $Y$ be positive dimensional irreducible projective varieties over $K$, let $f:X\dashrightarrow Y$ be a dominant…

Let $K$ be a global field and let $\mathscr{G}$ be a finite set of field generators. Consider $K$ as a structure over $\mathscr{L}=\mathscr{L}_a\cup\mathscr{G}$. Global-field rank…

Let $E/F$ be an extension of global fields, and let $\tau$ be as in the preceding notation. Write $N_{E/F}$ for the relative norm and let $\Lambda_E(s,\tau\circ N_{E/F})$ and…

Let $k$ be a number field, let $X$ and $Y$ be proper smooth geometrically integral varieties over $k$, and let $f:X\to Y$ be a dominant morphism. For each place $v$ of $k$, write…

Let $K$ be a global field, let $\mathcal{O}_S$ be a regular semilocal subring of $K$, and set $\mathcal{S}=\operatorname{Spec}(\mathcal{O}_S)$. Let $\mathcal{X}$ be a regular schem…

Global pullback functoriality conjecture. Every, or alternatively at least one, irreducible constituent of $\eta^*\pi$ has L-parameter $\Phi(\eta)\phi$. This is presented as a spec…

Counting conjecture. Both parts of the theorem stated for $n=2,3,4,5$ should hold for all $n$.

Let $K$ be a finite extension of $\mathbb{F}_p(t)$, let $\phi:\mathbb{P}^1\to\mathbb{P}^1$ be a rational map of degree $d\geq 2$ defined over $K$, and let $\overline{K}$ be the sep…

Let $K$ be a finite extension of the rational field $\mathbb{Q}$ or of the rational function field $\mathbb{F}_p(T)$. Let $(x_n)$ be the Newton approximation sequence associated to…

Schneider's conjecture. $i_n(F)=0$ for all $n\ne 1$. This predicts the vanishing of the divisible part of $H^2(G_S,W^n)$ outside the exceptional twist $n=1$; the supplied text give…

Let $F$ be a global field, and let $E(s,f,g)$ and $E(s,h,g)$ be the Eisenstein series in the source. For $F=\mathbb{Q}$, Corollary states that, for every $\varepsilon,\eta>0$, thes…