Analytification and tropicalization via hyperfields and definable types

Source abstract

We study the analytification and tropicalization of algebraic varieties over valued fields via Jun's spaces of hyperfield points. Our main new objects are a family of power-tropical hyperfields $H_n$ and their inverse limit $H$, recording at each level the class of a leading term modulo $n$-th powers of the residue field. They yield an enhanced tropicalization of the associated space $X(H)$, strictly finer than the Berkovich one exactly when the multiplicative group of the residue field is non-divisible: it specialises to the Berkovich analytification when the residue field is algebraically closed, to the real analytification of Jell-Scheiderer-Yu when it is real closed, and to a new $p$-adic analytification when it is $p$-adically closed. Over henselian valued fields of equicharacteristic $0$ we then construct a model-theoretic avatar of $X(H)$: the space $\widehat{X}_T$ of definable types orthogonal to the value group, which sharpens the geometry in the spirit of Hrushovski and Loeser and recovers their space over algebraically closed valued fields. We show that $\widehat{X}_T$ is strictly pro-definable, satisfies a model-theoretic Payne theorem, and definable compactness in such spaces satisfies a Heine-Borel type characterization. As an application, we analytify $p$-adic semi-algebraic sets and prove a strong density theorem for their tropicalizations, the exact $p$-adic counterpart of the real one of Jell-Scheiderer-Yu.