Points de torsion sur les varietes abeliennes de type GSp

TL;DR

This paper determines the optimal polynomial bounds for the number of torsion points over finite extensions on certain abelian varieties of GSp type, and explores implications for the Mumford-Tate conjecture.

Contribution

It computes the exact exponent for torsion point bounds on GSp-type abelian varieties and links the Mumford-Tate conjecture's validity for individual varieties to their products.

Findings

Optimal exponent for torsion bounds is established.

Unconditional results for varieties with trivial endomorphism ring outside an explicit set.

Conditional proof that the Mumford-Tate conjecture for individual varieties implies it for their products.

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 $\GSp$ type, i.e. whose Mumford-Tate group is "generic" (isomorphic to the group of symplectic similitudes) 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 $A$. The result is unconditional for a product of simple abelian varieties with endomorphism ring $\Z$ and dimension outside an explicit exceptional set $\mathcal{S}=\{4,10,16,32,...\}$. Furthermore, following a strategy of Serre, we also prove that if the Mumford-Tate conjecture is true for some abelian varieties of $\GSp$ type, it is then true for a product of such abelian varieties.