Hodge groups of Abelian varieties with purely multiplicative reduction

The main result of the paper is that if A is an Abelian variety over a subfield F of C, and A has purely multiplicative reduction at a discrete valuation of F, then the Hodge group of A is semisimple. Further, we give necessary and sufficient conditions for the Hodge group to be semisimple. We obtain bounds on certain torsion subgroups for Abelian varieties which do not have purely multiplicative reduction at a given discrete valuation, and so obtain bounds on torsion for Abelian varieties, defined over number fields, whose Hodge groups are not semisimple.