Geodesic
Hodge groups of abelian varieties with purely multiplicative reduction
Izvestiya. Mathematics , Tome 60 (1996) no. 2, pp. 379-389

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The main result of the paper is that if $A$ is an abelian variety over a subfield $F$ of $\mathbf C$, 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 therefore obtain bounds on torsion for abelian varieties, defined over number fields, whose Hodge groups are not semisimple. Bibliography: 26 titles. 
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     author = {A. Silverberg and Yu. G. Zarhin},
     title = {Hodge groups of abelian varieties with purely multiplicative reduction},
     journal = {Izvestiya. Mathematics },
     pages = {379--389},
     publisher = {mathdoc},
     volume = {60},
     number = {2},
     year = {1996},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1996_60_2_a5/}
}
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A. Silverberg; Yu. G. Zarhin. Hodge groups of abelian varieties with purely multiplicative reduction. Izvestiya. Mathematics , Tome 60 (1996) no. 2, pp. 379-389. http://geodesic.mathdoc.fr/item/IM2_1996_60_2_a5/