Torsion points on $\rm{GL}_2$-type abelian varieties

Abstract

It is well known that the rational torsion of an abelian variety defined over a number field injects into the reduction modulo any sufficiently large prime, so the order of the torsion group divides the greatest common divisor of the sizes of points on the reduction at each prime. Drawing inspiration from Katz's Inventiones paper (1981), we investigate the converse to this for abelian varieties of $\rm GL_2$-type and exhibit a conjectural list of possible torsion orders for modular abelian varieties over $\mathbb Q$ of dimension up to $5$.