A proof of a conjecture of Y. Morita

Abstract

A conjecture of Morita says that if an abelian variety defined over a number field has the Mumford-Tate group which does not have any non-trivial Q-rational unipotent element, then it has potentially good reduction everywhere. In this article, we prove this conjecture. The main ingredients of the proof include some newly established cases of the conjecture due to Vasiu, a generalization of a criterion of Paugam on good reduction of abelian varieties, and the local-global principle of isotropy for Mumford-Tate groups of abelian varieties.