Galois representations, Mumford-Tate groups and good reduction of abelian varieties

Abstract

Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent ℚ-rational points then A has potentially good reduction at any discrete place of K. The Mumford-Tate group is an object of analytical nature whereas having good reduction is an arithmetical notion, linked to the ramification of Galois representations. This conjecture has been proved by Morita for particular abelian varieties with many endomorphisms (called of PEL type). Noot obtained results for abelian varieties without nontrivial endomorphisms (Mumford’s example, not of PEL type). We give new results for abelian varieties not of PEL type.