UC Berkeley

The purpose of this thesis is to introduce and study the specialization map in the context of Scholze’s category of diamonds and to prove some basic results on its behavior. Our specialization map generalizes the classical specialization map that appears in the theory of formal schemes. Afterwards, as an example of interest, we study the specialization map for p-adic Beilinson-Drinfeld Grassmanians and moduli spaces of mixed-characteristic shtukas associated to reductive groups over Zp . Finally, as an application of our theory, we describe the geometric connected components of some moduli spaces of mixed-characteristic shtukas and local Shimura varieties at infinite level. This confirms and generalizes conjecture 4.26 of [46] in the unramified case.