Torsion pour les varietes abeliennes de type I et II

TL;DR

This paper determines the optimal polynomial bounds for torsion points of certain abelian varieties over number fields, extending known results and verifying the Mumford-Tate conjecture in new cases.

Contribution

It computes the best possible bounds for torsion points on abelian varieties of type I or II and proves the Mumford-Tate conjecture for new classes of these varieties.

Findings

Optimal bounds for torsion points are established.

The Mumford-Tate conjecture is verified in new cases.

Results are unconditional for certain abelian varieties of odd dimension.

Abstract

Let A be an abelian variety defined over a number field K, the number of torsion points rational over a finite extension L is bounded polynomially in terms of the degree [L : K]. When A is isogenous to a product of simple abelian varieties of type I or II in Albert classification and is "fully of Lefschetz type", i.e. whose Mumford-Tate group is the group of symplectic similitudes commuting with endomorphisms and which satisfy the Mumford-Tate conjecture, we compute the optimal exponent for this bound in terms of the dimensions of the abelian subvarieties of A and their rings of endomorphisms. The result is unconditional for a product of simple abelian varieties of type I or II with odd relative dimension. Extending work of Serre, Pink and Hall, we also prove that the Mumford-Tate conjecture is true for a few new cases for such abelian varieties.