Etale subquotients of prime torsion of abelian schemes

Abstract

Let $A$ be an abelian variety over a number field $K$ with good reduction outside a finite set of primes $S$. We show that if the $\ell$-torsion subgroup schemes $A[{\ell }^n]$ lie in a certain category of group schemes, then $A[{\ell }^n]$ does not contain any subgroup schemes that are étale or are of multiplicative type.