Endomorphism rings of reductions of elliptic curves and Abelian varieties

Let E be an elliptic curve without CM that is defined over a number field K. For all but finitely many non-Archimedean places v of K there is a reduction E(v) of E at v that is an elliptic curve over the residue field k(v) at v. The set of v's with ordinary E(v) has density 1 (Serre). For such v the endomorphism ring End(E(v)) of E(v) is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers N and M there are infinitely many non-Archimedean places v of K such that the discriminant Δ(v) of End(E(v)) is divisible by N and the ratio Δ(v)/N is relatively prime to NM. We also discuss similar questions for reductions of Abelian varieties. The subject of this paper was inspired by an exercise in Serre's "Abelian l-adic representations and elliptic curves" and questions of Mihran Papikian and Alina Cojocaru.