On the \(\ell\)-adic Galois representations attached to nonsimple abelian varieties (Q332218)

Let \(A_1\) and \(A_2\) be abelian varieties defined over a field which is finitely generated over rationals and \(\ell\) be a prime number. For any abelian variety \(A\) the group \(H_\ell(A)\) is the connected component containing zero of the intersection of the algebraic monodromy group at \(\ell\) with the general linear group of the rational Tate module at \(\ell\). The first result of the paper is a sufficient criterion for \(H_\ell(A_1 \times A_2)\) to be isomorphic to \(H_\ell(A_1) \times H_\ell(A_2)\). The conditions are difficult to check in general, but as a consequence one gets that the Mumford-Tate conjecture holds for all abelian variety of any dimension which are isogenous to a product of abelian varieties of dimensions 1 and 2. It is also shown that a similar result in positive characteristic is true, if some additional technical hypotheses are added.