Lower bounds for ranks of Mumford-Tate groups

Details Lower bounds for ranks of Mumford-Tate groups Lower bounds for ranks of Mumford-Tate groups

2011-10-31

arxiv.org/abs/1110.6816v1

Mathematics

Algebraic Geometry

Scientific paper

Let A be a complex abelian variety and G its Mumford--Tate group. Supposing that the simple abelian subvarieties of A are pairwise non-isogenous, we find a lower bound for the rank of G, which is a little less than log_2 dim A. If we suppose further that End A is commutative, then we show that rk G >= log_2 g + 2, and this latter bound is sharp. We also obtain the same results for the rank of the l-adic monodromy group of an abelian variety defined over a number field. ----- Soit A une vari\'et\'e ab\'elienne complexe et G son groupe de Mumford--Tate. En admettant que les sous vari\'et\'es ab\'eliennes simples de A sont deux \`a deux non-isog\`enes, en trouve une minoration du rang de G, un peu moins que log_2 dim A. Si on suppose en plus que End A soit commutatif, alors on montre que rk G >= log_2 g + 2, et cette borne-ci est la meilleure possible. On obtient les m\^emes resultats pour le rang du groupe de monodromie l-adique d'une vari\'et\'e ab\'elienne d\'efinie sur un corps de nombres.

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