Néron models of pseudo-Abelian varieties

Abstract

We study Néron models of pseudo-Abelian varieties over excellent discrete valuation rings of equal characteristic $p>0$ and generalise the notions of good reduction and semi-Abelian reduction to such algebraic groups. We prove that the well-known representation-theoretic criteria for good and semi-Abelian reduction due to Néron–Ogg–Shafarevich and Grothendieck carry over to the pseudo-Abelian case, and give examples to show that our results are the best possible in most cases. Finally, we study the order of the group scheme of connected components of the Néron model in the pseudo-Abelian case. Our method is able to control the $\ell$-part (for $\ell \ne p$) of this order completely, and we study the $p$-part in a particular (but still reasonably general) situation.