Torsion for abelian varieties of type III

TL;DR

This paper determines the optimal polynomial bounds for the number of rational torsion points on certain abelian varieties over number fields, under specific geometric and conjectural conditions, and proves the Mumford-Tate conjecture in new cases.

Contribution

It computes the exact exponent for torsion bounds for abelian varieties of type III under specific conditions and proves the Mumford-Tate conjecture for new cases of Lefschetz type varieties.

Findings

Optimal torsion bounds are established for abelian varieties of type III.

The Mumford-Tate conjecture is proven for new classes of abelian varieties of Lefschetz type.

The bounds depend on the dimension of subvarieties and their endomorphism rings.

Abstract

Let $A$ be an abelian variety defined over a number field $K$. The number of torsion points that are rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$ of $L$ over $K$. Under the following three conditions, we compute the optimal exponent for this bound, in terms of the dimension of abelian subvarieties and their endomorphism rings. The three hypothesis are the following: $(1)$ $A$ is geometrically isogenous to a product of simple abelian varieties of type I, II or III, according to the Albert classification; $(2)$ $A$ is of "Lefschetz type", that is, the Mumford-Tate group is the group of symplectic or orthogonal similitudes which commute with the endomorphism ring; $(3)$ $A$ satisfies the Mumford-Tate conjecture. This result is unconditional for a product of simple abelian varieties of type I, II or III with specific relative dimensions. Further, building on work of Serre, Pink, Banaszak, Gajda and Kraso\'n, we also prove the Mumford-Tate conjecture for a few new cases of abelian varieties of Lefschetz type.