Lower bounds for ranks of Mumford-Tate groups

TL;DR

This paper establishes lower bounds for the rank of Mumford--Tate groups of complex abelian varieties, relating it to the dimension and endomorphism properties, with sharp bounds proven.

Contribution

It provides new lower bounds for the rank of Mumford--Tate groups based on the abelian variety's dimension and endomorphism structure, including sharp bounds.

Findings

Lower bound for Mumford--Tate group rank is just below log_2 of the dimension.

If End A is commutative, the rank is at least log_2 of the dimension plus 2.

Results extend to the l-adic monodromy group of abelian varieties over number fields.

Abstract

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 that End A is commutative, then we show that rk G >= log_2 dim A + 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 supposant que les sous vari\'et\'es ab\'eliennes simples de A sont deux \`a deux non-isog\`enes, on trouve une minoration du rang rk G de G, l\'eg\`erement inf\'erieure \`a log_2 dim A. Si on suppose que End A est commutatif, alors on montre que rk G >= log_2 dim A + 2, et cette borne-ci est optimale. 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.