Greenberg functors and jumps of tori

This thesis comprises two parts, both pertaining to arithmetic geometry. In the first one we give a formal axiomatisation of Greenberg functors; to this end, we introduce and study a notion of geometric category which abstracts the properties of the category of schemes that are relevant for Greenberg’s construction, allowing us to treat other categories, like various subcategories of that of logarithmic schemes, at the same time. In this context, we study the representability of functors “à-la Greenberg”, as well as their preservations of some properties like quasi-compactness or quasi-separatedness. We then apply this formalism both to examples already studied in the literature and to new ones: among these, we develop and study logarithmic versions of the Weil restriction and of the classical mixed-characteristic Greenberg functor. The second part mainly deals with Edixhoven’s jumps and Chai’s conductors of tori, which are invariants related to the behaviour of their Néron models under base-change. The main results concern Edixhoven’s jumps of induced tori over an arbitrary discretely valued strictly henselian field K and, more generally, of K-tori which are direct factors of K-rational varieties.