6 problems

Let $K$ be a global function field of characteristic different from $p$, with absolute Galois group $G_K$. Let $n$ be a positive integer and let $\mathfrak{R}$ be an effective divi…

Let $\mathbb{F}_s$ be the algebraic closure of $\mathbb{F}_q$ in $K_v$, let $z$ be a uniformizer, and identify $K_v=\mathbb{F}_s((z))$ and $A_v=\mathbb{F}_s[[z]]$. Let…

Let $v$ be a non-archimedean place of a number field $F$, let $k_v$ be the residue field with $q_v=\#k_v$, and let $\pi_v:X_v\to B_v$ be the reduction of a Fano fibration. Let…

Let $R$ be an excellent henselian discrete valuation ring with residue field $k$, and let $B=\operatorname{Spec}R$. Let $\mathcal Y$ be a regular scheme, flat and proper over $B$,…

Let $K$ be a global function field over a finite field $k$ of positive characteristic, let $S$ be a finite set of places of $K$, and let $\Gamma\subset O_S^*$ be a subgroup. Fix a…

Existence conjecture. It may be conjectured that there are infinitely many pairwise non-isomorphic elliptic curves $E/F$ and infinitely many imaginary quadratic field extensions…