On the arithmetic surjectivity of morphisms

Contact: Pieter Moree (moree@mpim-bonn.mpg.de)

Let $k$ be a number field and $k_v$ denote the completions of $k$ at its places $v$.~Recall that a morphism $f:X\to Y$ of $k$-varieties is called arithmetically surjective (a.s.) if $f$ is surjective on the $k_v$-rational points for almost all places $v$.  Supposing the $X,Y$ are proper and smooth, the Colliot-Thélène conjecture (CCT, 2008) gives conjecturally a sufficient condition on $f$ to be a.s.
The CCT was proved by Denef (2019) in an even sharper form, and Loughran-Skorobogatov-Smeets (2020) gave a full characterization of a.s. morphisms in terms of pseudo-splittnes. The plan for my talk is: First, review the "old'' results; second, present (wide) generalizations of some of those results, e.g. by allowing $k$ to be finitely generated, ${\rm char}(k)=0$; finally, say a few words about my approach, which is new and also shows that a.s. is equivalent to completely birational conditions on proper morphisms $f$, provided $X,Y$ are "sufficiently'' smooth. Time permitting, I will comment on the case of positive characteristic, where the question is open.