Algebraic groups associated with abelian varieties
The author extends the known results of Tankeev, Ribet, and the author on the Hodge groups and Hodge cycles of simple abelian varieties (over the field of complex numbers) to the case of non-simple abelian varieties. Especially, he shows that all Hodge cycles on non-simple abelian varieties (containing simple factors of type I, II, III, and IV) are generated by Hodge cycles of degree \( 2\) and canonical Hodge cycles constructed by Murty, under certain condition on the dimensions of those simple factors. Moreover, he shows similar results on the Zariski closures of the \(\ell\)-adic Galois groups with respect to the \(\ell\)-adic Tate modules of abelian varieties over \(\ell\)-adic local fields.
\(\ell -adic\) Tate modules of abelian varieties over \(\ell \)-adic local fields
scientific article; zbMATH DE number 4142208
scientific article