5 problems

Genus finiteness conjecture. The set ${\bf gen}(G)$ is finite.

Let $K$ be a finitely generated field over its prime field, and let Statements (a)--(c) in (3) of the theorem be the three assertions referenced in the source. Geyer–Jarden conject…

Height gap conjecture. There exists some $\alpha\in\widehat{\operatorname{Pic}}(k_1)_{\operatorname{int}}$ with positive arithmetic degree such that

Let $K$ be a field. For an absolutely almost simple simply connected algebraic $K$-group $G$, write $\mathbf{gen}_K(G)$ for its genus, the set of $K$-isomorphism classes of $K$-for…

Finite generation conjecture. If $M$ is PAC over $K$, then