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

TL;DR

This paper explores the relationship between torsion points on $ m{GL}_2$-type abelian varieties over number fields and their reductions, proposing a conjectural classification of torsion orders for certain modular abelian varieties.

Contribution

It investigates the converse of known torsion injection results for $ m{GL}_2$-type abelian varieties and presents a conjectural list of possible torsion orders for low-dimensional cases.

Findings

Established a conjectural list of torsion orders for modular abelian varieties up to dimension 5.

Connected torsion point behavior with reductions modulo primes in the context of $ m{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 $5$.