Points de torsion sur les varietes abeliennes de type GSp

Details Points de torsion sur les varietes abeliennes de type GSp Points de torsion sur les varietes abeliennes de type GSp

2009-11-29

arxiv.org/abs/0911.5505v2

Mathematics

Number Theory

31 pages, new section 5, accepted for publication in Journal de l'Institut de Mathematiques de Jussieu

Scientific paper

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.

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