Etale subquotients of prime torsion of abelian schemes

TL;DR

This paper investigates the structure of prime torsion subgroup schemes of abelian varieties over number fields, showing restrictions on their subgroup schemes based on certain categorical conditions.

Contribution

It establishes that under specific categorical assumptions, the prime torsion subgroup schemes of abelian varieties lack étale or multiplicative subgroup schemes.

Findings

Prime torsion subgroup schemes do not contain étale subgroup schemes.

Prime torsion subgroup schemes do not contain multiplicative subgroup schemes.

Results apply to abelian varieties with good reduction outside a finite set of primes.

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 \'etale or are of multiplicative type.