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The main result of the paper is that if $A$ is an abelian variety over asubfield $F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at adiscrete valuation of $F$, then the Hodge group of $A$ is semisimple. Further,we give necessary and sufficient conditions for the Hodge group to besemisimple. We obtain bounds on certain torsion subgroups for abelian varietieswhich do not have purely multiplicative reduction at a given discretevaluation, and therefore obtain bounds on torsion for abelian varieties,defined over number fields, whose Hodge groups are not semisimple.Comment: This is an updated version of the paper. LaTeX2e or LaTeX2.09 or  AMSLaTeX. Contact: silver@math.ohio-state.ed

Hodge groups of Abelian varieties with purely multiplicative reduction