Bounds for torsion on abelian varieties with integral moduli

TL;DR

This paper establishes bounds on torsion points of abelian varieties with integral moduli over local and global fields, and applies these bounds to classify torsion structures in specific cases.

Contribution

It introduces a function bounding torsion subgroup sizes for abelian varieties with potentially good reduction, and applies this to classify torsion in CM elliptic curves and abelian surfaces.

Findings

Bound on torsion subgroup size for abelian varieties with good reduction

Classification of torsion points on CM elliptic curves over degree ≤3 fields

Identification of 11 possible torsion orders for abelian surfaces over Q

Abstract

We give a function F(d,n,p) such that if K/Q_p is a degree n field extension and A/K is a d-dimensional abelian variety with potentially good reduction, then #A(K)[tors] is at most F(d,n,p). Separate attention is given to the prime-to-p torsion and to the case of purely additive reduction. These latter bounds are applied to classify rational torsion on CM elliptic curves over number fields of degree at most 3, on elliptic curves over Q with integral j (recovering a theorem of Frey), and on abelian surfaces over Q with integral moduli. In the last case, our efforts leave us with 11 numbers which may, or may not, arise as the order of the full torsion subgroup. The largest such number is 72.