Abeloid varieties over higher rank valued fields: Raynaud uniformization and Néron models

Abstract

We study abeloid varieties over the adic space $\operatorname{Spa}(K, K^+)$, where $K$ is a complete non-Archimedean field and $K^+$ is a valuation ring of arbitrary rank. We extend the well-known Raynaud uniformization to this setting. We also show the existence of suitable Néron models in algebraic and formal situations over higher rank valuation rings: We prove the existence of algebraic (lft-)Néron models for semi-abelian varieties over valuation rings with algebraically closed fraction fields, as well as, up to finite separable base change, over complete rank one valuation rings with algebraically closed residue field. Building on this result, we show the existence of formal marked Néron models for abeloid varieties over $\operatorname{Spa}(K, K^+)$ after finite separable base change. We also investigate the connection between the higher rank Raynaud extension and the formal Néron model.