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 nonarchimedean places $v$ of $K$ there is the 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 nonarchimedean places $v$ of $K$ such that the discriminant $\mathrm{\Delta }(v)$ of $End(E(v))$ is divisible by $N$ and the ratio $\mathrm{\Delta }(v)\mathrm{/}N$ 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 $\mathrm{\ell }$-adic representations and elliptic curves” and questions of Mihran Papikian and Alina Cojocaru.