Existence of global Néron models beyond semi-abelian varieties
Otto Overkamp, Takashi Suzuki
TL;DR
This work resolves the global existence problem for Néron models in positive characteristic by proving the BLR90 conjectures when residue fields are perfect, while producing a counterexample in the imperfect case using wound unipotent groups and their duality via relatively perfect unipotent duals. The authors introduce and leverage the framework of relative perfection, particularly the duality against the Hodge–Witt–type sheaves $ u_n(1)^{RP}$, to construct relatively perfect Néron models and perform a dévissage that reduces complex cases to two fundamental ones. They establish a precise criterion distinguishing when Néron models exist and are quasi-compact, and they demonstrate non-existence in the imperfect case through explicit analysis of $ u(r)$ with $r>1$, aided by Cartier operators. Finally, they classify unirational wound unipotent groups up to relative perfection in the perfect residue-field setting, showing a canonical equivalence with $p$-primary finite étale group schemes and identifying key correspondences with Jacobians and Néron-model phenomena. The results provide both negative answers in general and a robust positive theory in the perfect-residue setting, with concrete applications to Tamagawa numbers and arithmetic of function fields.
Abstract
We first prove Bosch-Lütkebohmert-Raynaud's conjectures on existence of global Néron models of not necessarily semi-abelian algebraic groups in the perfect residue fields case. We then give a counterexample to the existence in the imperfect residue fields case. Finally, as a complement to the conjectures, we classify unirational wound unipotent groups "up to relative perfection", again in the perfect residue fields case. The key ingredient for all these is the duality for relatively perfect unipotent groups.